LIBRARY 

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University  of  California. 

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Class 

I.  The  Syzygetic  Pencil  of  Cables  with  a  new  Geometrical 

Development  of  its  Hesse  Group,  Gg,^. 

II.  The  Complete  Pappus  Hexagon. 


DISSERTATION 


SUBMITTED)    TO   THE   BOARD   OF   UNIVERSITY   STUDIES   OF  THE  JOHNS    HOPKINS  UNIVERSITY  IN 
CONFORMITY  WITH  THE  REQUIREMENTS  FOR  THE    DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


BY 

CHARLES  CLAYTON  GROVE 


BALTIMORE,  MD. 

June,  1906. 


I.  The  Syzygetic  Pencil  of  Cnbics  with  a  new  Geometrical 

Development  of  its  Hesse  Group,  Ggig. 

II.  The  Complete  Pappns  Hexagon. 


DISSERTATION 


SUBMITTED  TO  THE  BOARD  OF  UNIVERSITY  STUDIES  OF  THE  JOHNS   HOPKINS  UNIVERSITY  IN 
CONFORMITY  WITH  THE  REQUIREMENTS  FOR  THE   DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


BY 

CHARLES  CLAYTON  GROVE 


BALTIMORE,  MD. 

June,  1906. 


Z^t  Jgorb  (gaftttnore  qpnee 

BALTIMORE,  MD.,  U.  S.  A. 

1907 


,^ 


INTRODUCTION. 


1.  The  course  of  lectures  by  Prof.  Frank  Morley  during  the  winter  of 
1903-4  on  cubic  curves  suggested  this  dissertation  and  prepared  me  to  carry  on 
the  research.  The  trend  was  largely  determined  by  an  incidental  question  by 
Prof.  A.  Cohen  as  to  the  groups  involved  in  the  system  of  conies  which  I  had 
just  presented  to  the  Mathematical  Seminary.  The  interest  and  valuable  sug. 
gestions  of  Dr.  A.  B.  Coble  in  the  carrying  on  of  the  work  are  gratefully 
acknowledged. 

2.  The  close  connection  between  the  Hesse  group  and  the  syzygetic  pencil 
of  cubics  makes  it  necessary  to  say  at  least  something  about  this  pencil  of  curves. 
Without  attempting  even  an  outline  of  the  theory,  I  present  in  Section  I.  only 
such  matter  as  is  needed  later,  besides  some  new  facts  concerning  the  pencil  and 
a  figure  showing  the  appearance  of  some  noteworthy  and  specially  related  cubics 
of  the  pencil.  No  figure  seems  ever  to  have  been  published  except  that  in  con- 
nection with  the  paper  of  Prof.  Morley  in  the  Proceedings  of  the  London  Math- 
Society,  Ser.  2,  Vol.  2,  Part  2,  which  shows  arbitrarily  selected  cubics.  The 
initial  and  all  but  the  closing  work  leading  to  that  figure  was  done  by  me. 
Therefore,  I  present  a  figure  of  the  pencil  herein,  also  one  of  the  corresponding 
polar-reciprocal  range  of  line  cubics. 

Section  II.  shows  how  to  derive  a  closed  system  of  thirty-six  conies  analogous 
to  the  conic  of  Section  I.  as  to  which  the  pencil  and  range  are  polar-reciprocal. 
It  also  discusses  the  action  of  the  polarities  of  these  conies  upon  the  four 
inflexional  triangles,  and  presents  some  history  of  similar  considerations. 

In  Section  III.  there  is  given  a  brief  history  of  the  attempts  to  determine  all 
finite  groups  of  transformations,  and  in  particular  an  account  of  the  Hesse  Group 


165264 


4  Introduction. 

of  216  collineations.  Further,  we  derive  and  write  down  the  matrices  of  these 
collineations  by  means  of  the  closed  system  of  thirty-six  conies,  which  are  here 
differently  defined  than  in  Section  II.  and  are  given  accordingly.  All  the  sub- 
groups are  found  and  discussed.  The  collineations  are  finally  classified  as  to 
periodicity. 

Section  IV.  treats  of  triangles  in  other. perspective  forms  than  six-fold  as  are 
the  inflectional  triangles  above.  As  a  second  way  to  secure  triangles  in  three- 
fold perspective,  also  some  in  two-  and  one-fold  perspective,  we  develop  what 
we  call  the  Complete  Pappus  Hexagon,  in  its  dualistic  forms  and  deduce  a 
number  of  theorems  connected  with  it. 


I. 

THE  SYZYGETIC  PENCIL.    DRAWINGS  OF  THE  PENCIL  AND  RANGE. 


1.     The  name  stzygetic^  is  given  to  the  pencil  of  cubics  determined  by  a 

cubic/  and  its  Hessian  A, 

x/4-;iA  =  0,  (1) 

which  has  the  same  nine  inflexions  for  all  its  cubics,  the  intersections  of  the  two 
cubics /and  A,  and  so  has  the  same  four  inflexional  triangles. 

It  is  well  known  that  any  non-singular  cubic  may  be  brought  in  four  ways, 
as  shown  for  example  by  Weber,  ^  into  Hesse's  canonical  form 

xl-\- xl-\- xl-\- 6mxiXciXs=  0.  (2) 

This  simply  means  that  the  cubic  has  been  referred  to  one  of  its  inflexional 
triangles  as  reference  triangle.     The  Hessian  covariant  of  the  form  (2)  is 

A  =  —  m^xf  +  xl-^xl)-^{l  ■^2m^)x,X2Xs.  (3) 

Its  vanishing  gives  the  Hesse  Cubic  or  Hessian.     Thus  if  m  is  the  parameter  of 

the  cubic  (2)  and  m'  is  that  of  its  Hessian,  we  have  6  m'  =  — ^ — ;  and 

form  (2)  for  all  values  of  m   from   —  oo  to  +  <»    gives  the   pencil   as   well 

as  form  (1). 

The  relation  between  w  and  m'  shows  that  each  cubic  of  ike  pencil  has  hut 

one  Hessian  hut  is  Hessian  to  three  cubics  of  the  pencil. 

Choosing  one  of  the  inflexional  triangles  as  reference  triangle,  we  readily 

calculate  in  turn, 

The  coordinates  of  the  nine  inflexions, 

Their  arrangement  on  the  sides  of  each  of  four  triangles,  ^ 

The  equations  of  the  sides  and  opposite  vertices  of  these  triangles, 

1  Clebsch-Lindemann :  Lecons  sur  la  Geometric,  II,  p.  230. 
"Lehrbuch  der  Algebra,  3.  Aufl.,  II,  §§  106,  107;  ss.  399-404. 
8  C.-L. :  II,  (7)  p.  233. 


6  I.     The  Syzygetic  Pencil.     Drawings  of  the  Pencil  and  Range. 

The  polar  conies  of  the  inflexions,  which  are  in  each  case^  two  right  lines, 

viz.,  the  inflexional  tangent  and  the  harmonic  polar  of  the  inflexion. 
For  future  reference  we  give  the  sides  of  the  inflexional  triangles : 
A.  B.  C.  D. 

jCj  =  0.     £Ci  -f-  a^g  +  ajg  =  0.  o^  Xi  -f-  ccg  +  iCg  :=  0.     co  aji  -f-  ajg  +  aJg  =  0. 

052=0.       a^i  +  a>^iC2  H-G)iC3=  0.        JCj  +  C0^iC2+  JC3=  0.       «!  -f  G)  ^2  +  iCs  =  0-      (4) 

Xg  =  0.     jcj  +  G)  ir2  H-  o^ iCa  =  0.      iCj  +  a;2  +  "^  s-g  =  0.     iCi  +  a^  +  o)  JC3  =  0. 

The  equations  of  the  vertices  respectively  opposite  are  given  by  exchanging 
^  for  X  and  interchanging  a  and  o^.  a  and  o^  are  the  complex  cube  roots 
of  unity. 

2.     Some  Particular  Cubics  of  the  Pencil. 

(a)  The  cubics  whose  parameter  m  is  respectively  00,  — ^,  —  \a?^  — ^co 
are  the  four  inflexional  triangles  in  the  order  as  obtained  above,  of  which  ^  two 
are  real  and  two  imaginary.  The  polar  line^  as  to  these  triangles  of  any  pointy 
has  coordinates  respectively, 


«! 

^ 

X3 

A: 

^2^8          ' 

Vzyi 

yiy2' 

B: 

y\  —  yzyz 

yl     y%y\ 

yl  —  yiy2' 

C: 

y\  —  ^^y%yz 

y\  —  ^^yzy\ 

yl  —  ^^yiy^ 

D: 

yl  —  ^y^y-i 

yl  —  (^yzy\ 

y\  —  ^yxyi' 

The  determinant  formed  from  any  three  rows  of  these  coefficients  vanishes 
identically,  therefore  the  four  polars  pass  through  a  common  point,  so  we  say. 

The  four  polar  lines  of  any  point  as  to  the  four  inflexional  triangles  meet  in  a 
point,  or  also 

Any  two  of  the  inflexional  triangles  are  apolar  as  seen  from  any  point  of  the 
plane,  and  are  thus  syzygetic. 

{h)  The  first  of  two  covariant  cubics  of  the  pencil  is  the  Hessian,  equation 
(3),  p.  5.  The  second  is  the  Cayleyan  of  the  cubic  (2).  The  polar  conic  as  to 
this  cubic  of  a  point  y  is 

yy  {x\  +  2 wa-ga-g)  -f  ^g (a^l  +  2  w x^x^  -^^  y^[x\-\-  2m x^ x^  =  0. 
Considering  the  y's  as  parameters,  this  is  a  net  of  polar  conies.   By  inspection, 
we  see  that  the  conic  w^j  —  ^2^8  is  apolar  with  the  three  conies  of  the  net; 

>C.-L.:  II,  p.  227;  Salmon:  Higher  Plane  Curves,  8.  Ed.,  §§  74,  170,  pp.  69,  146. 
«  C.-L. :  II,  p.  230,  also  pp.  239,  310. 
»H,  P.  C,  8165,  p.  143. 


I.     The  Syzygetic  Pencil.     Drawings  of  the  Pencil  and  Range.  7 

likewise  the  conies  m^l  —  ^3^1,  and  m^l — ^1^3.  So  the  net,  has  a  corre- 
sponding weh  of  apular  conies,  given  by  the  equation 

where  the  jy's  are  parameters. 

The  contravariant  or  Jacobian  of  the  web  is  the  Cayleyan : 

2  w  ^1         —  ^3  —  ^2 

-^3  2771^3  —^1  =0, 

—  ^2  —^1  2  771^3 

or  wa?  +  ^i  +  ^i)  +  (l-4m^)^i^3^3  =  0, 

which  is  a  cubic  of  the  range^  enveloped  by  the  oo^  lines  composing  the  degenerate 

conies,  the  first  polars  of  points  along  the  Hessian  of  the  cubic. 

Here  as  in  the  case  of  the  Hessian,  we  see  that  each  cubic  of  the  range  has 
but  one  Cayleyan  but  it  is  Cayleyan  to  three  cubics. 

(c)  The  simplest  invariant  is  found  by  operating  with  the  Cayleyan  on  the 
cubic.  ^  It  is  the  quartic  invariant  denoted  in  Salmon  by  S.  By  operating 
thus  and  dividing^  by  24  we  have 

S  =  m{l  —  m^). 

The  vanishing  of  this  invariant  gives  the  parameters  of  four  curves  of  the  pencil 
called  the  equianharmonic  cubics : 

jS^:  xl-\-  xl+  xl  =  0.  Ssi  xl-{-xl  +  xl  + 60^x^x^X3=  0. 

^Sg :  ccf  +  x|  +  aj|  -f-  6  Xi  ccg  2*3  =  0.     S^:  a:?  +  a^l  +  xi  +  6  0  Xj  Xg  ccg  =  0. 

This  name  is  given  because  the  constant  anharmonic  ratio  of  the  four  tangents* 
drawn  from  a  point  of  the  curve  tangent  to  the  curve  itself  is  equianharmonic 
in  these  four  cases. 

(d)  By  operating  with  the  Cayleyan  on  the  Hessian  we  have  the  sextio 
INVARIANT,  T  of  Salmon.^     It  is 

T=l  —  20m^—8m^. 

iC.-L.:  II,  p.  344. 

2 By  operating  with  a  line-form,  as  of  the  Cayleyan,  upon  a  point-form,  as  of  the  cubic  (2)  called/,  we 
mean  that  the  f's  are  taken  as  partial  differential  operators.  The  equation  of  the  Cayleyan  means  in  this 
process 


3 Salmon:  H.  P.  C,  §220,  p.  191. 

*C.— L.:  II,  pp.  325,  326. 

5H.  P.  C,  §221,  p.  193,  also  Weber:  II,  s.  406  (7). 


8  I.     The  Syzygetic  Pencil.     Draxmngs  of  the  Pencil  and  Range. 

The  curves  whose  parameters  are  given  for  7=0,  are  the  six  harmonic  cubics 
of  the  pencil,  called  so  since  the  ratio  of  the  four  tangents  to  the  curve  from 
points  on  it  is  harmonic.  ^ 

With  this  amount  of  introduction  and  number  of  references  necessary  to 
prepare  the  student  to  read  the  whole  most  profitably,  and  with  the  equations 
at  hand  to  which  reference  must  later  be  made,  we  pass  now  to 

The  Drawing  of  the  Syzygetic  Pencil. 

3.  As  a  far  more  convenient  form  of  the  cubics  of  the  pencil  for  purposes 
of  construction,  we  transform  the  equation  of  the  pencil  so  as  to  have  inflexions 
at  the  circular  imaginary  points  /  and  /,  by  putting 

aa  =  a;  +  y  +  1,         x^  =  —  {x—\),         xs  =  —  (y—l),  (6) 

where  x  and  y  are  conjugate  coordinates. 

By  this  transformation,  equation  (2)  of  the  pencil  becomes 

or  it  is  of  the  form 

xy{x-\-y)  +  (i{a^-\-xy-\-y')-{-l=  0,  (7) 

,  2(1— m),  2  —  u 

where  u  =  /  .    ^ — <- ,  and  m  =  ^  .     ,  ^  ,  . 

^        1  +  2m  '  2(1  +  ^) 

4.  The  Special  Cubics  of  §  2. 

(a)  The  four  inflexional  triangles  (4)  become  in  conjugate  coordinates, 

A:  {x  +  y+l){x  —  l)(y-l)  =  0. 

B:  a?  +  xy-\-f  —  (x  —  ciy){x  —  (J'y)  =  0. 

C:  xy{x-{-y)-(^ia^-^xy'hf)-\-l  =  0.  ^    ^ 

D:  xy{x-^-y)  —  ci^3^-{-xy-\-y^)  +  l  =  0. 

Our  former  reference  triangle  A  is  seen  thus  to  be  now  the  line  through  the 
points  G)  and  o^  on  the  unit-circle,  and  the  point  x=  1  taken  twice. 

Since  the  cube  terms  are  lacking  in  B,  the  cubic  consists  of  the  line  at 
infinity,  also,  as  readily  seen,  of  the  lines  through  the  origin  and  o  and  u>^ 
respectively.  The  representable  parts  are  shown  in  the  figure  of  the  pencil. 
The  three  dash  lines  are  the  real  harmonic  polars,  the  full  lines  are  the 
degenerate  cubics. 

»  C.-L. :  II,  p.  326. 


I.      The  Syzygetic  Pencil.     Drawings  of  the  Pencil  and  Range, 
(h)  The  Hessian  by  direct  calculation  on  (7)  is 


1 

2y-{-  2(1 

2{x  +  y)-^li 

(i{2x  +  y) 

6 

2(a;  +  2/)+i^ 

2x-\-  2^ 

ii{2y-\-x) 

li{2x  +  y) 

(i{2y-{-x) 

6 

or  E=^ii'xy{x  +  y)  —  (^^^-^){a?-^xy-\-y'')^Zii^z=0^  (9) 

which  is  of  the  form  of  the  cubic  (7)  with  its  parameter 

^'givenby^'  =  — ^^-ti. 

Therefore  equation  (7)  likewise  gives  the  syzygetic  pencil  for  all  values  of  ^ 
from  — CO  to  +  00  as  well  as  equation  (1). 

The  Cayleyan  is  calculated  as  follows : 

The  polar  conic  as  to  the  cubic  (7)  of  a  pointy  is 

Regarding  the  jp's  as  parameters  this  is  a  net  of  polar  conies. 
To  find  the  weh  of  apolar  conies  we  take  the  general  line-conic 

«f  +  hr^  +  c^  +  d^n  +  en^  +/f  ^  =  0, 

and  operate  separately  on  the  point  conies  of  the  net.     Thus  we  obtain 

25+  2c?+  2^/-(-  ^e  =  0, 
2a  +  2d-\-  y.f+  2^e  =  0, 
2(ia  +  2(ih  -\-  ^d  +  ec  =  0. 

We  desire  apolar  conies  analogous  in  form  respectively  to  those  of  the  net  and 
so  we  first  take  a  =  0,  b=zd=zl^  whence,  by  substituting  in  the  three  equations 
above  and  in  the  general  line-conic,  we  get  the  conic 

2(1^71+  2fiy!  —  ^^  —  [jp=z0. 

Second,  put  5  =  0,  a  =  c?  =  1,  and  we  get  the  conic 

2fi^^+2iz^yi^4yj—fi^  =  0. 

Third,  put  a  =  l,e=/=0,  and  the  corresponding  conic  is 

2(P-^>7H->7')-i^  =  0. 

By  these  three  we  may  write  the  apolar  web  of  line-conics. 

Hi l2(z^yj+2fiyi'-4^-i/]  +n,[2^f  +  2fi^yj-4y!-(,^  -f  n3[2 {^^-^yj+yj^)-^-]  =  0. 


10 


I.     The  Syzygetic  Pencil,     Drawings  of  the  Pencil  and  Range. 


The  Jacobian  of  this  web  is  by  direct  calculation 

tin— 2  /t*(^+2>7)         —(2^ +  ^2) 

2^  — >7  — ^4-2>7  —ii 

or  2^(f+>7^)-3^^,7(^+>7)H-(^^-2)(f-^,7+>7')-i^'(^->7)-i^=0,     (10) 

which  is  the  Cayleyan  of  the  cubic  (7). 

(c)  By  operating  with  the  Cayleyan  (10)  on  the  cubic  (7)  we  have  the 
qnartic  invariant  o 4 j, 

whose  vanishing  gives  the  parameters  of  the  four  equianharmonic  cubics  of  the 
pencil,  viz.,  /tz  =  0,  2,  2{o,  2(a^  That  two  of  these  are  real  is  seen  from  the 
parameters. 

(d)  By  operating  with  the  Cayleyan  on  the  Hessian  (9)  we  obtain  the 

sextic  invariant 

T=S  —  20(1^  —  fi\ 

whose  vanishing  gives  the  parameters  of  the  six  harmonic  cubics,  viz.,  the  roots 
of  |it^  =  —  10  ±  6  Vs.  For  the  positive  and  the  negative  sign,  there  are  in  each 
case  one  real  and  two  imaginary  roots,  or  there  are  two  systems  of  three  values 
each  of  the  harmonic  ratio.     The  two  real  roots  are 

—  (1  —  V3)  and  —  (1  +  V3). 

5.  The  nine  inflexions  and  harmonic  polars  mentioned  in  §1,  pp.  5,  6  are 
in  conjugate  coordinates  as  follows,  with  the  positions  on  the  figure  as  indicated 
respectively  — 

Inflexions.  Point  on  figure.    Its  Harmonic  Polar. 
1.        At  00  on  6)0)^  X  —  y=0 


2. 

/ 

3. 

J 

4. 

a' 

6. 

6. 

7. 

a 

8. 

9. 

The  Line  on  the  Figure. 

Axis  of  reals 

Circular  imaginary  rays 

from  the  origin 


Line  1,  — a^ 


x+  2y  =  0\ 

2x  +  y  =  0} 
x  —  cd^y  +  uP  —  1  =0 
X  —  u^y  —  o  4-1  =  0     A  cir.  imag.  ray  through  o 

X O^y  —  6)^  +  G)  =  0 

X  —  iny  +0  —  1  =  0 

X 


6) 


Line  1,  —  q 
0)//  —  (j2_|_i=o     A  cir.  imag.  ray  through  o^ 


X  —  wy  —  a)4-(o^  =  0 


<^   . 


(11) 


I.     The  Syzygetic  Pencil.     Drawings  of  the  Pencil  and  Range.  11 

By  projecting  the  pencil  into  this  form,  we  have  the  line  at  infinity  as  one 
real  side  of  an  inflexional  triangle,  and  one  real  inflexion  at  infinity.  That 
leaves^  to  appear  on  the  figure  three  sides  of  inflexional  triangles,  two  inflexions 
and  three  harmonic  polars. 

Towards  a  better  understanding  of  the  syzygetic  pencil  and  the  drawing  of  it 
we  analyze  it  further  and  present  first  amongst  our  findings. 

6.  The  Asymptotes. 

Take  the  line  a:  +  y  =  X,  which  is  perpendicular  to  the  axis  of  reals  at  a 
distance  ^7^  from  the  origin;  that  is,  the  line  in  which  the  reflexion  of  the  origin 
is  the  point  7..    It  cuts  the  cubic  (7)  where 

Xx(a  — a)  +  /^(a;2  — XiB  +  X^)  +  1  =  0. 

Since  the  cube  terms  vanish,  the  line  meets  the  cubic  at  infinity.   If  next  we  put 

X  =  /[i,  we  have  only 

^3+1  =  0. 

Hence,  since  the  square  terms  also  vanish,  the  line  x  -\-  y  =  ^  is  tangent  to  the 
cubic  at  infinity,  in  direction  perpendicular  to  the  axis  of  reals  and  is  the 
asymptote. 

The  point  at  infinity  on  the  line  uuF,  perpendicular  to  the  axis  of  reals  is  as 
above  noted  one  of  the  real  inflexions,  hence  the  asymptote  just  found  is  an 
inflexional  tangent  as  well. 

Thus  when  ^i  is  given,  the  asymptote  of  the  cubic  of  the  pencil  for  that 
particular  ^  as  parameter  is  also  given  as  the  perpendicular  to  the  axis  of  reals 
at  the  distance  ^^  from  the  origin. 

7.  By  taking  the  first  polar  of  the  inflexion  q  as  to  the  general  cubic  (7) 
we  find  it  breaks  up  into  the  two  linear  factors 

\x  —  G)Z/-f-t^  —  If  jw^(ic  —  uiy  —  6)-|-  \)  ■\-  ^  {x  —  o^2/)[  =^  ^) 
the  harmonic  polar  and  flex-tangent  respectively. 

The  latter  cuts  the  asymptote  of  the  cubic  where  l/^ 

—  (1  +  |ti)(G)x  +  oV  — "^+  1)  =  0. 
The  harmonic  polar  of  uf'  [see  equations  (11),  p.  10]  cuts  the  asymptote 
where  —  (g)£c  +  g)^^  —  o^  -f-  1)  =  0.     Therefore,  we  may  draw  the  flex-tangents 
to  any  cubic  of  pencil  hy  drawing  from  either  inflexion,  6>  or  u^,  to  the  point  where 
the  harmonic  polar  of  the  other,  w^  or  o,  cuts  the  asymptote  of  that  cubic. 

1  C.-L.  :  II,  p.  235  on  the  real  parts  of  the  figure. 


12  T.     The  Syzygetic  Pencil.     Drawings  of  the  Pencil  and  Range. 

8.  Intersections  of  the  Cubics  with  the  Axis  of  Reals. 
The  general  cubic  (7)  cuts  the  axis  of  reals  where 

2ic^+  3|Ma^4-  1  =  0. 

The  discriminant  (Weber:  I,  s.  273)  of  this  equation  is 

D  =  —  108(^^4-  1), 
which  shows  that  if 

1.  ft  <;  —  1,  then  Z>  ]>  0,  and  the  equation  has  three  real  distinct  roots. 

2.  /u  =  —  1,  then  D  =  0,    "      *^  "         "    one  real  repeated  root. 

3.  fi'p-  —  1,  then  i>  <:^  0,    "      "  "         "    one  real,  two  imaginary  roots. 

As  to  the  cubics  this  says  that 

1.  foTfK^  —  1,  the  cubic  cuts  the  axis  of  reals  in  three  points  and  hence  is, 
in  general,  of  the  bipartite  type. 

2.  for  /^  =  —  1,  there  is  one  actual  intersection  and  an  acnode  on  the  axis, 
as  was  noted  in  §  4  (a),  p.  8. 

3.  for  ^  >-  —  1,  there  is  but  one  real  intersection  and  the  cubic  is  of  the 
unipartite  type. 

The  study  of  these  intersections  thus  enables  us  to  classify  the  cubics  from 
the  parameter. 

9.  Besides  the  aids  in  constructing  the  syzygetic  pencil  furnished  by  these 
facts  as  to  the  asymptotes,  the  inflexional  tangents,  and  the  intersections  of  the 
cubics  with  the  axis  of  reals,  we  present,  as  the  final  means  of  facilitating  the 
construction,  the  drawing  of  a  number  of  circles  concentric  with  the  unit-circle 
and  the  calculation  of  the  intersections  of  the  cubics  with  these  circles. 

The  cubic  (7)  cuts  the  circle  xy  =  p^  where 

Since  x  and  y  are  conjugate  complex  coordinates,  a:  +  «-  =  c  is  a  right  line 

perpendicular  to  the  axis  of  reals  at  a  distance  ^  c  from  the  origin.     Therefore, 
the  cubic  cuts  the  circle  where  this  perpendicular  does,  for  values  of  c  which  are 

the  roots  of  the  equation  in  fic  +  ^  J  . 

The  location  of  the  perpendicular  is  given  by 

4fi  ^       ' 


I.     The  Syzygetic  Pencil.     Drawings  of  the  Pencil  and  Range.  13 

10.  Two  interesting  matters  of  analysis  will  be  noted  before  presenting 
the  drawing.  The  Hessian  cubic  whose  parameter  is  ^i'  is  Hessian  to  three  cubics 
whose  parameters  are  the  roots  of 

^3  +  3^^3  +  4=0.  (13) 

Their  asymptotes  cut  the  axis  of  reals  at  i  /[^  resp.  The  Hessian  itself  cuts  the 
axis  of  reals  at  points  given  by  the  roots  of  2a;3  +  S/tz'cc^  +  1  =  0. 

If  we  put  in  this  equation  x=-\^i,  we  obtain  identically  the  former 
equation.  Therefore,  the  Hessian  cuts  the  axis  where  the  asymptotes  of  its 
curves  do,  and  these  asymptotes  are  its  tangents  on  the  axis  of  reals.  Or,  for 
this  form  of  the  pencil,  we  may  say,  the  Hessian  is  tangent  on  the  axis  of  reals  to 
the  asymptotes  of  its  curves. 

Since  the  axis  of  reals  is  the  harmonic  polar  of  the  real  inflexion  at  infinity 
and  the  asymptotes  are  flex-tangents  at  this  inflexion,  we  may  state  the  theorem 
projectively : 

The  three  flex-tangents  of  the  three  cuhics  with  a  common  Hessian,  at  any  one 
inflexion,  touch  this  Hessian  in  three  points  on  the  harmonic  polar  of  the  inflexion 
considered. 

This  is  an  extension  of  theorems  by  Alfred  Clebsch^  and  Peter  Muth.^ 

The  discriminant  of  equation  (13)  is  Z>  =  —  432  (^'^  +  1).  Thus,  if  the 
Hessian  is  bipartite,  i.  e.,  if /u'<]  —  1,  there  are  three  real  unipartite  cuhics  of  which 
it  is  Hessian;  for  the  roots  of  equation  (13)  are  then  all  real,  two  positive  and 
one  negative  but  greater  than  —  1. 

If  (i'  =  —  1,  its  cubic  is  Hessian  of  the  cubics  whose  parameters  are 
—  1,  2,  2.  That  is,  it  is  Hessian  of  itself  and  of  one  of  the  two  real  harmonic 
cubics  twice  over. 

If  the  Hessian  is  unipartite  it  is  the  Hessian  of  one  real,  bipartite  cubic. 

Since  our  names  unipartite  and  bipartite,  following  Salmon,  are  actually 
names  of  the  cubics  von  Staudt  calls  resp.  odd-  and  even  circuit  cubics,  and 
since 3  by  no  projection  does  a  non-singular  cubic  change  its  class  of  odd-  or  even- 
circuit,  these  facts  as  to  the  Hessian  and  its  cubics  remain  for  all  real  projections. 

11.  The  two  real  harmonic  cubics  are  readily  seen  to  be  mutually  Hessian 
and  cubic,  for  the  parameters  —  (1  +  V3)  and  —  (1  —  \/3)  may  be  interchange- 
ably (J,  and  fi'  and  satisfy  equation  (13). 

1  Ueber  die  Wendetangenten  der  Curven  dritter  Ordnung ;  Crelle  Journal  58,  s.  232. 

^Ucber  ternare  Formen,  u,  s.  w. ;  Inaug.  Diss Giessen,  1890,  s.  15. 

SC.-L. :  II,  end  of  foot-note  p.  223. 


14  I.     The  Syzygetic  Pencil.     Drawings  of  the  Pencil  and  Range. 

Therefore,  the  flex-tangents  of  each  touches  the  other  and  so,  in  our  form, 
each  cuts  the  axis  of  reals  at  the  asymptote  of  the  other.  Compare  Clebsch  in 
Crelle  Journal,  Bd.  58,  ss.  238,  239. 

12.  For  the  form  (7)  of  the  pencil,  the  two  invariants  are  got  very  nicely 
as  the  invariants  of  the  quartic  ^  giving  the  intersections  of  the  four  tangents 
from  the  real  inflexion  at  infinity. 

As  shown  in  §  6,  p.  11,  the  flex-tangent  at  infinity  cuts  the  axis  at  a;  =  ^jw, 
and  in  §8,  p.  12,  the  cubic  cuts  the  axis  at  2ic^  +  3//a:^  -f-  1  =  0.  Therefore,  the 
intersections  of  the  four  tangents  from  the  real  inflexion  at  infinity  are  given  by 
the  quartic  (2  a^  +  3/1^0^  -|-  1)  (2x  — -  /i^)  =  0,  or 

4ic*  +  4^a^  — 3^^0:2 -f  2a;  — /[i  =  0. 

The  invariants^  of  this  quartic  are  those  of  the  cubic,  given  in  §  4,  (c)  and  (cZ), 
to  within  a  numerical  factor. 

The  Syzygetic  Range  of  Cubics.^ 

13.  The  invariant  parts  of  the  syzygetic  pencil  just  studied  are  well  known 
and  are  known  to  correspond  dualistically  throughout.  In  view  of  this  corre- 
spondence it  is  observed  that  the  polarity  arising  from  the  conic 

a;SH-«i  +  «i=o,  (14) 

namely,  «i  =  ^i,         3^2  =  ^2,         «3  =  ^8, 

sends  each  part  into  its  corresponding  part,  also  the  point-cubics  of  the  pencil 

into  the  line-cubics  of  range, 

^f+e  +  ^i+6m^i^3J3  =  0.  (15) 

The  conic  (14)  transformed  into  conjugate  coordinates  becomes 

2(a^-f  a;2/  +  2/^)  +  3  =  0, 

which  equation  shows  the  conic  to  be  an  hyperbola  with  the  inflexional  lines 
(see  -B,  (8),  p.  8)  x  —  (ay  =  0  and  x  —  id^y  =  0  as  asymptotes,  with  vertices  at 
±:  hi^/Q.  Transformed  to  rectangular  coordinates  with  the  axis  of  reals  as  the 
X-axis,  it  becomes  6X^  —  2  F^  -|-  3  =  0;  and  with  the  asymptotes  as  axes  it  is 
XF= — '.  From  this  equation  we  easily  construct  the  conic  by  considering 
the  equality  of  segments  of  chords  contained  between  the  curve  and  its 
asymptotes. 

1  Salmon:  H.  P.  C,  §228,  p.  199. 

«  Weber:  I,  $  70,  8.  230;  II,  §  108,  s.  406. 

•C.-L.:  II,  p.  244  sq. 


I 

Fig.  1.— The  Syzy'getic  Pencil  of  Cubics. 


Fig.  2.  — The  Syzygetlc  Range,  polar  reciprocal  of  Pencil  as  to  Conic 


II.     On  a  Closed  System  of  Conies.  15 

By  the  polar- reciprocal  process  as  to  this  conic  we  deduce  the  line-cubics  of 
the  syzygetic  range  from  the  point-cubics  of  the  pencil.  The  harmonic  polars 
are  seen  to  become  the  cusp-tangents  common  to  all  the  curves  of  the  range. 

Explanation  op  the  Figures. 

The  Syzygetic  Pencil.  The  inflexional  lines  are  solid.  The  harmonic 
polars  are  dash  lines.  All  cubics  are  marked  with  their  parameters.  Specially 
related  ones  are  the  same  kind  of  lines. 

The  cubics  represented  are : 

1.  Two  equianharmonic,  ^  ^  0  and  2. 

2.  Two  harmonic,  ii=.  —  (1  —  V^S)  and  —  (1  +  VS).  They  cut  the  axis 
of  reals  respectively  at  —  i  (1  +  VS)  and  —  J  (1  — V3);  the  former  is  uni- 
partite,  the  latter  bipartite.  Their  inflexional  tangents  from  u  and  o^  are  drawn 
in  fine  dotted  lines. 

3.  The  cubic  for  which  /[^  =  —  §  is  Hessian  of  three  for  which  ^  is  resp.  4, 
1.132f,  —  0.88 2^.     These  four  are  drawn  alike. 

4.  The  cubic  ^  =:  ^  is  Hessian  of  but  one  real  cubic  for  which  ^^=^  —  2. 

5.  The  equianharmonic  cubic,  ff  =  0,  is  also  Hessian  of  but  one  real  cubic 
for  which  ^  =  — \/4.     It  is  draw  different  from  the  others. 

The  Syzygetic  Range.  The  majority  of  the  above  cubics  were  reciprocated 
in  the  conic  shown  on  the  figure  and  the  polar-reciprocal  line-cubics  appear 
drawn  in  the  same  kind  of  lines  as  the  corresponding  point-cubics  from  which 
they  were  derived  and  are  named  with  the  same  value  of  ^  respectively. 

Clebsch  has  a  drawing  (C.-L.:  II,  p.  243)  of  the  range  but  does  not  profess 
it  to  have  been  constructed.  Its  form  is  very  considerably  diff*erent  from  the 
accompanying  drawing. 

II. 

ON  A  CLOSED  SYSTEM  OF  CONICS. 

14.  The  conic  (14)  of  §  13,  p.  14,  is  referred  to  triangle  A  of  equations  (4), 
p.  6,  as  reference  triangle  with  the  first  line  of  B  (ibid.)  as  auxiliary  line.  Its 
effect  on  the  parts  of  the  syzygetic  pencil  as  given  in  §  13,  raised  the  question  as 
to  the  effect  of  the  analogous  conies  as  to  all  the  possible  reference  frames  in  the 
four  triangles  (4). 

The  equations  of  these  analogous  conies  may  be  got  by  considering  separately 
each  of  the  four  triangles  as  reference  triangle  with  the  nine  remaining  lines  in 


16 


II.     On  a  Closed   System  of  Conies. 


turn  as  the  auxiliary  line.  Thus,  there  are  thirty-six  such  conies,  nine  for  each 
triangle.  As  an  example,  say  we  wish  to  consider  triangle  B  and  the  first  side 
of  triangle  G  as  reference  frame.     Then, 

Sci  =  &)  a^i  +  Ci)  ^2  +  "  ^3  =  0, 

x'2  =  oci    +  o^a-g  4- oajg  =  0, 

Xs  =  Xi     -\-  (dX2  +  (0^ Xg  =z  0 , 

whence  x[  -\-  X2  -\-  Xs  =  (o^  +2  6))  (cj^ccj  +  iCg  +  a^)  =  0  is  auxiliary  line  as  we 
desired.  The  conic  x[^  -f-  x^'^  -f  jcg^  =  0  has  therefore  as  to  the  original  reference 
frame  the  equation 

a  xl  -i-  xl  -^  xl  -\-  2  {0x^X3  -^  XqXi  -\-  X1X2)  =  0 . 

Similarly  by  inspection  all  the  others  may  be  deduced. 

The  equations  of  these  conies  will  be  given  in  the  next  chapter  where  they 
are  derived  differently  and  their  properties  are  defined.  Their  development  as 
above  was  published  in  The  Johns  Hopkins  University  Circular,  January  1905, 
pp.  16  ff. 

These  thirty-six  conies  form  a  closed  system  for  by  operating  two  times  with 
a  polarity  arising  from  any  conic  of  the  system  we  get  a  polarity  of  the  system ; 
or,  the  product  of  three  polarities  of  the  system  is  a  polarity  of  the  system. 

It  is  easily  shown  that  when  a  conic  reciprocates  a  triangle  into  another, 
the  two  triangles  are  in  perspective,  and  conversely.  For  two  triangles  in  n-fold 
perspective  there  are  n  such  conies,  so  for  the  four  inflexional  triangles,  mutually 
in  six-fold  perspective,  there  should  be  six  times  ^Gz  or  thirty-six  conies, 
as  there  are. 

15.  As  to  their  effect  upon  the  inflexional  triangles,  the  polarities  divide 
into  sets  in  two  ways.  First,  by  nines  Ai_q,  5i_9,  etc.,  they  send  the  vertices 
respectively  of  triangles  A,  B,  etc.,  into  the  sides  opposite,  and  at  the  same  time 
reciprocate  another  triangle  into  itself  and  the  vertices  of  each  of  the  other  two 
triangles  into  the  sides  of  the  other  of  these  two  triangles.  Second,  they  divide 
into  sets  of  six  each,  operating  as  indicated  in  the  table : 

Polarities.  Triangles.  Triangles. 

into  themselves 
send  A  and  B  respectively,  and  G  and  D  into  each  other 


-^1-8  J 

B^_^  sen 

Ai-ij 

^4-6 

A^-9, 

A-3         *' 

^l-B, 

^7-9 

A-6, 

■B.-.     " 

^1-3, 

A-»    " 

A 

"    G 

A 

"    D 

B 

''    G 

D 

"    B 

G 

"    B 

D 

B 

B 

G 

A 

D 

A 

G 

A 

D 

II.     On  a  Closed  System  of  Conies.  17 

The  classification  shows  also  that  besides  the  nine  polarities  which  send  a 
triangle  into  itself,  vertices  into  sides  opposite,  there  are  nine  others  sending  it 
into  itself,  but  vertices  into  sides  in  anti-cyclic  order  as  to  the  order  of  the  vertices. 

We  shall  speak  of  the  three  cyclic  and  the  three  anti-cyclic  forms  of  per- 
spective, meaning  as  indicated  here. 

Cyclic  Forms.  Anti-cyclic  Forms. 

V         2'         Z'  '                   V         2'         Z' 

12          3  13          2 

2  3          1  2          13 

3  12  3          2          1 

As  analysis  readily  shows,  the  projecting  rays  are  the  nine  harmonic  polars 
with  four  vertices  of  the  triangles  on  each.  The  centers  of  perspective  are  the 
vertices  of  the  triangles,  and  the  axes  their  sides.  Further,  for  any  two  of  the 
triangles,  the  vertices  and  sides  of  one  of  the  remaining  two  triangles  are 
respectively  the  centers  and  axes  in  cyclic  order  for  the  three  cyclic  per- 
spectivities,  and  those  of  the  other  triangle  respectively  in  anti-cyclic  order  are 
centers  and  axes  for  the  three  anti-cyclic  perspectivities. 

16.  "We  would  call  attention  to  two  very  neat  papers  on  perspective  triangles 
by  J.  Valyi.^  He  makes  a  slight  error  as  to  6-fold  perspective  triangles  by  saying, 
"Unter  den  6  Kegelschnitten  giebt  es  hochstens  vier  reelle,  die  beiden  Dreiecke 
sind  immer  imaginar,"  whereas  our  equations  (4),  p.  6,  show  one  whole  triangle 
and  one  side  of  the  other  may  be  real. 

After  publishing  the  article  in  the  J.  H.  U.  Circular  (§  14,  p.  16),  I  found 
the  following  papers  by  S.  Kantor. 

In  1895  he  notes ^  the  36  collineations,  in  connection  with  two  triangles  in 
6-fold  perspective.  The  following  year  he  speaks^  of  36  conies  in  connection 
with  the  four  Hesse  Triangles.  He  uses  these  conies,  which  are  the  36  herein 
presented  but  defined  differently  as  stated  in  Th.  XIII  and  used  to  operate  on 
the  collineations  of  types  5,  6,  7  of  Jordan  *  to  produce  the  most  general  group 
of  correlations  which  contain  these  types. 

By  finding  the  "intermediate^^  or  Salmon's  contravariant  conic  <I)^  of  each 

1  Archiv  der  Mathematik  und  Physik:  1882,  Bd.  70,  ss.  105-110;  1884,  2.  R.,  II.  T.,  ss.  230-234. 

^Theorie  der  endlichen  Gruppen  von  eindeutigen  Transformationen  in  der  Ebene,  (Berlin)  a.  58. 

»Crelle  Journal,  Bd.  116,  ss.  176,  177. 

*Ibid.,  Bd.  84,  s.  92. 

5  Salmon-Fiedler :  Kegs,  6.  Aufl.,  II,  s.  668. 


I 


18  III.      The  Hesse  Group  0/  216   CoUineations. 

two  conies  in  the  36  and  remembering  the  geometrical  meaning  of  the  inter- 
mediate, we  learn  the  relative  positions  of  the  36  conies.  Also,  this  process 
gives  us  other  sets  of  line-eonies  mostly  of  27  each  but  none  are  the  27  having 
six-fold  contact  with  a  cubic  of  the  pencil  ^  as  we  hoped.  These  intermediates 
are  not  sufficiently  pertinent  to  our  subject  to  be  given  here. 

111. 

THE  HESSE  GROUP  OF  216  COLLINEATIONS. 

17.     History  and  Bibliography. 

The  attempts  to  determine  all  finite  groups  of  transformations  run  back  to  the 
work  of  F.  Klein  in  the  number  for  July  1874  of  the  Sitzungsherichte  der  Erlanger 
phys.-med.Gesellschaft :  Ueber  binare  Formen.  Again  in  1876,  Math.  Ann.  9:185, 
he  sets  the  problem,  '*Alle  Gruppen  anzugeben,  welehe  aus  einer  endlichen  An- 
zahl  von  linearen  Transformation  en  bestehen,"  and  proceeds  to  determine  them  by 
the  method  of  rotations  of  the  regular  polyhedrons  into  themselves  respectively. 

In  1876,  L.  Fuchs  in  Crelle  Journal  81:97-142,  and  the  following  year 
P.  Gordon  in  Math.  Ann.  12 :  23-46,  confirm  the  results  of  Klein  for  the  binary 
domain  by  entirely  different  methods. 

Camille  Jordan  in  a  notable  memoir  (1878,  Crelle  Journal,  84:89-215) 
tried  to  give  completely  the  groups  of  both  the  binary  and  the  ternary  domain. 
He  discovered  the  group  6^210,  mentioned  ibid.  p.  206  and  called  the  Hesse  group. 
He  had  noted  its  existence  before  the  writing  here  cited  as  he  therein  states. 
Two  years  later  (1880),  Jordan  devotes  himself  directly  to  the  group  problem 
in  a  memoir  in  the  Atti  della  R.  Accademia  d.  Scienze  Fisiche  e  Math.  Societa 
Reale  di  Napoli,  Vol.  VIII,  No.  11,  pp.  1-41.  It  is  worthy  of  note  that  Jordan 
did  not  find  the  simple  group  (xjeg,  discovered  by  Klein  [Math.  Ann.  (1879) 
14:428-471]. 

Next,  in  1887,  the  Hessian  group  is  considered  by  Alexander  Witting  in  hie 
inaug.  diss.  (Gottingen,  58,  S.  8)  by  the  use  of  hyperelliptic  functions. 

Heinrich  Maschke  gives  the  fullest  treatment  of  the  subject.  See  Math. 
Ann.  29:157  ff . ;  Nachrichten  d.  K.  Gesellsehaft  d.  Wiss.  zu  Gottingen,  (1888) 
Nr.  5,  ss.  78  ff. ;  and  especially  in  connection  with  his  presentation  of  the  group 
of  51840  transformations  (Math.  Ann.  33  :  317-344)  in  1889. 

«lbid.:  Hohern  Eb.  Knr.,  8166,  8.  178. 


III.      The  Hesse  Group  0/  216   CoUineations.  19 

Again  in  1889  we  have  an  important  memoir  by  H.Valentiner:  ''De  endlige 
Transformations -Gruppers  Theori,"  in  Copenhagen-  K.  Danske  Videnskabs. 
Selskab.,  Naturvid.  og  Math.  6.  Raekke  V.  :  2,  pp.  67-204;  French  resum6 
pp.  205-235.  Valentiner  shows  rather  clearly  that  he  did  not  know  of  the 
memoire  of  Jordan  and  on  pp.  151,  222,  denies  the  existence  of  the  group  of 
216  collineations.  He  demonstrates  (p.  69)  the  possibility  of  a  group  of  72 
transformations  containing  ones  of  2nd,  3rd  and  4th  order,  where  the  trans- 
formations of  the  4th  order  by  sixes  have  a  common  second  power.  Later  he 
shows  that  the  group  really  exists.  He  discovered  the  group  G^^  and  presents 
it  on  pp.  191-198,  French  pp.  231-233. 

A.  Wieman  refers  to  the  Hessian  group  in  connection  with  his  paper  in 
Math.  Ann.  47:531-556:  "  Ueber  eine  einfache  Gruppe  von  360  ebenen 
Collineationen,"  (1896),  and  of  course  in  his  article  on  ''Endliche  Gruppen 
Linearer  Substitutionen"  (1900)  in  Encyklopadie  der  Mathematischen  Wissen- 
schaften,  Bd.  I,  H.  5,  3  f.     (See  p.  528.) 

S.  Kantor's  paper  referred  to  in  §  16,  p.  17,  though  written  in  1896,  makes 
no  provision  for  nor  mention  of  G-seo  discovered  at  least  seven  years  before. 

In  the  Kansas  University  Quarterly  of  January  1901,  H.  B.  Newson 
presents  ''The  Group  of  216  Collineations  in  the  Plane,"  from  the  one  to  eighteen 
correspondence  with  the  tetrahedral  group.  This  paper  was  not  known  to  me 
when  my  note  was  made  in  the  J.  H.  U.  Circular  of  Jan.  1905. 

18.  Relative  to  the  Hesse  group  the  purpose  is  to  derive  its  collineations 
by  purely  geometrical  processes  from  certain  conies. 

Theorem.  Three  point-conics  such  that  each  is  apolar  to  the  remaining  two  in 
line  form  give  rise  to  and  determine  the  syzygetic  pencil  of  cubics  in  the  Hesse 
canonical  form . 

The  three  conies  a?  +  2  w XgXg  =  0, 

xl+  2mxsXi=0,  (16) 

xl  +  2mxiX2  =  0, 

are  of  this  sort ;  and  it  has  been  shown  that  in  the  net  of  conies  determined  by 
three  arbitrary  conies  there  are  just  four  sets  of  three  conies  each  of  this  type, 
and  that  those  of  each  set  are  tangent  at  the  vertices  by  twos  to  the  sides  of  one 
of  the  four  inflexional  triangles  of  the  cubic  of  the  net  of  conies. 

The  Jacobian^  of  these  forms  (16)  is  the  cubic  curve,  the  locus  of  points 

iC.-L. :  I,  p.  378. 


20  III.     The  Hesse  Group  0/  216   ColUneaiions. 

whose  polars  as  to  the  three  conies  meet  in  a  point,  which  point  is  also  on  the 
cubic  and  has  the  former  point  as  its  correspondent  in  the  same  way  as  it  is  of 
that  point.     The  cubic /is  then 


/  = 


^1 

mxs 

771x2 

mxQ 

Xz 

mxi 

mx2 

mxi 

3*3 

=  0, 


or  /=  — m2(x?  +  ic|  +  a:|)  + (1  -f  2m3)xiCC2X3  =  0,  (3) 

which  is  the  Hessian  of  the  cubic  (2)  §  1  whose  parameter  is  6  m,  and  so  is  a 
member  of  the  syzygetic  pencil  given  by  equation  (2)  for  all  positive  and 
negative  values  of  m  from  0  to  00 .  Therefore,  by  varying  the  parameter  in 
equations  (16)  we  have  three  similar  pencils  of  conies  for  the  vertices  two  and 
two  which  give  rise  to  the  syzygetic  pencil  of  cubics  as  above  stated. 

19.     The  Thirty-six  Conics. 

The  inflexional  triangles  of  this  pencil  are  given  by  equations  (4)  and  the 
four  equianharmonic  cubics  have  equations  (5)  p.  7  to  define  them. 

The  thirty-six  conics  are  the  first  pilars  of  the  vertices  of  the  inflexional  triangles 
as  to  the  four  equianharmonic  cubics,  less  twelve  degenerate  conics  which  are  the 
squares  of  the  sides  severally  of  the  inflexional  triangles. 

These  twelve  come  from  taking  polars  as  indicated  hy  Ai Si,  B^Sz,  O^Ss, 
DiSi,  (i  =:  1,  2,  3).  The  subscript  i  indicates  the  vertex  opposite  the  sides  in 
order  as  given  in  equations  (4),  the  letters  show  the  respective  triangles  and 
equianharmonic  cubics. 

The  conics  in  this  way  are  as  follows: 

B,8r.  x'i-^xl  +  xl  =  0.  0,S,:  o)xl-\- xl-\-xl=zO.        D.S^:  a)'xl-^xl-^xl=zO. 

BiSy:  x\-\-wxl  +  w^xl  =  0.        0^8^:  xl-\- o}xl-\-xl  =  0.        D^S,:  xl-\- (o^xl  +  xl=zO. 

BfiS,:  xl-\-w^xl^o)xl  =  0.       C^S,:  xl-\-xl-\- wxl  =  0.        D,S,:  x\-[-xl-\- <o'xl  =  0. 

G1S2:   a)xl-\-xl-{-xl-\-2{a)X2Xs-{-XsXi-\-XxX-^)=.0. 

C282 :   xl-^(oxl-\-xl-\-  2{x2Xs-^  (oXiXi-^  Xi x^  =z  0. 

O3S2 :   xl-^xl-\-(oxl-\-2 {X2 Tg  +  3^3 «!  4-  a> Xi X2)  =  0. 

D1S2:    a)'^x\-\-xl-^xl-\-2{w^X2Xs-{-XiXi-\-XiX2)  =  0. 

D2S2:   xl-\-a)^xl-\-xl-{-  2{x2X3-\-(o'^x,iXy-^XiX2)  =  0. 

DaSi :   xl-{-xl-^Q)^xl-\-2 (jCj x^  -\-XiXi-\-  o? x^x^  =  0. 

A-y8i\  x\-\-2x2X^z=.(i. 

A2S2:   tcl -|- 2 a^g a;i  =  0. 

A3S2:  xl-{- 2x1X2=:  0. 


III.     The  Hesse  Group  of  216    Collineations.  21 

D1S3'.  a/ Xi -\- xl -{•  xl -{- 2  o)^  {co^  Xi  x^  +  a^s «!  -}-  a^i  a^a)  ==  0- 
DiS^:  xl -}-  io^ xl -{- xl -{- 2  0)^ {x2 x^  +  co^ x^ x^  -\-  x^ X2)  =  0. 
1)3/^3:   xl-\-xl-\-  o)^ ajg  +  2  ft>^  {x2  a*3  +  a^g  cci  +  «>^  ^i  ^2)  =  0. 

-4i /S'3 :   x\  -\-  2  0/  X2X3=z  0. 

AiSzi  a^H- 2w^a;3a;i  =  0. 

AsSs'.   »!  -f~  2  <o^  a?!  a;2  =  0. 

-Si/8'3:   a;?4- a;H-a;l-J- 2w2(a;2a;3  +  a;3a;i  +  ^i^3)  =  0- 

^2 'S^s  5   »!  -J-  a>  a^l  4"  ^^  ^H~  2  o>^  (a^a  a^s  +  w  a^s a^i  +  w^  x^  x-i)  =  0. 

BiS^:  x\ -\-  0)^ xl -\-  CD xl  •\- 2 o? [x^ x^  -\-  o? XzX-i-\-  m x^ a^)  =  0. 

Ax^i'.   a;? -f-  2  a>  a^a a^a  =  0. 

A<iSC'   a;2  + 2a>a?sa;i  =  0. 

A^H^'.   a;3-|- 2tt>a;ia;2:=  0. 

-Si 'S'4 :  x\-^x\-\-x\-\;-2oi (ajg x^  +  a^s a^i  +  ^1  ^2)  =  0. 

BzSi'.  a;J-|- wa;!  4-^^2;3-j- 2w(a;2a;3  + coa;3a;i4- w^a^ia^g)  =  0. 

B^Sii  xl -{-  0)^  xl -\-  a)  xl -\- 2  0)  {x2  Xs  -\-  0)^X3  Xi-{-  (o  x^  x^  =  0. 

C\ ^4 :  o)x\  -^  x\-\-  xl  -\-  2 (1)  {(o XiXz-\-  XzXx-\- XiX.^  =z  0. 

(72/8'4:  a;i4-(wa;2  +  a;3  4- 2ft>(a;2a;3  4-<wa;3a;i4-a;ia;2)  =  0. 

CzS^:  «?  + a;^  + wa;|  +  2w(a;2a;3  +  a;3a;i-|-a>a;ia;2)  =  0. 

These  conies  form  a  closed  system  as  explained  and  are  in  the  same  order 
except  ^2^3  ^^^  BiS^  as  if  derived  as  in  §  14.  Further,  the  line-forms  are  given 
by  interchanging  o  and  o^  and  writing  ^^  for  x^.  We  name  point-conics  with 
Roman  caps;  the  same  letters  in  script  name  the  corresponding  line-conics. 

20.    The  Conics  as  Source  op  the  216  Collineations. 

The  product  of  a  point-conic  on  a  line-conic  is  a  collineation.  With  this  in 
mind  we  form  a  multiplication  table  with  the  point-conics  along  the  left  side 
and  the  line-conics  along  the  top.  The  216  collineations  are  written  in  matrix 
form  and  are  numbered.  The  number  is  put  in  the  square  on  the  table  corre- 
sponding with  the  two  conics  which  produce  the  collineation  of  that  number. 
Thus  arranged  the  collineations  were  readily  classified,  from  their  actions  on  the 
inflexional  triangles  (and  for  simplicity  on  triangle  A)  into  the  tetrahedral 
subgroups. 

In  the  multiplication  of  matrices  remember  that  the  separate  terms  of  the 
ROWS  UPON  the  terms  of  the  columns  of  multiplicand  give  rows  of  product. 

In  numbering  I  put  ajh,  where  a  is  the  number  of  the  collineation  given 
and  h  is  that  of  the  one  having  g)  and  a^  interchanged. 


22 


III.      The  Hesse  Group  o/  216   CoIIineations. 


21.    The  216  Collineations. 


1 
0 
1 
0 
10 


79/94 
10    0 
0    0     1 
0    w2   0 

85/103 


2/3 
0 

CO 

0 
11 


4 
1 

0 

0 

12 


10    0 

10    0 

0     10 

0    0     1 

0    0     1 

0     10 

0     1     0 

0     CO    0 

0     0     1 

10    0 

o^'^  0     0 

0     0    CO 

0    0     1 

0     0  co"" 

10     0 

0     10 

0     CO    0 

a>2   0     0 

5 

0 

0 

1 

13/16 


10    0 

0    0     1 

0     10 

10    0 

0    0     1 

0     10 

10    0 

0     0    CO 

0     10 

10    0 

0    0     1 

0    ft>3   0 

6/9 
0 

'  0 

CO 

14/17 
0    0     1 
0     CO    0 
co^   0     0 


73/91 

74/92 

75/93 

co^  0     0 

0    0     1 

0     10 

0     0    1 

0      6>2     0 

1     0   0 

0     10 

1     0    0 

0    0   w^ 

80/95 
0     0   a.2 

0     10 
10    0 

86/104 


81/96 
0     10 
co^  0    0 
0    0     1 

87/105 


76/97 
10    0 
0     0   co^ 
0     10 

82/JOO 
a>2   0     0 
0     10 
0     0     1 

88/106 


77/98 
0    0     1 
0     10 
w2   0     0 

83/101 
10    0 
0    a>'    0 
0     0     1 

89/107 


7/8 

1 

0 

0 

15/18 


0     10 

CO    0     0 

0     0   co^ 

19/28 
111 

20/29 
co^  1    a 

21/30 

CO^     CO      1 

22/34 
111 

23/35 
CO     co^    1 

24/36 

\     O?    CO 

1    co^  a) 

111 

CO     cu^    1 

CO      1     (O^ 

1   1   1 

\     CO      CI? 

1      CO      Cl? 

CO     1    co^ 

111 

CO^    1     CO 

CO^    CO       1 

Ill 

25/31 
111 

26/32 
1     CO     co^ 

27/33 
CO     1    co^ 

O?     CO      \ 

111 

CO^    1     CO 

0)      CI?    \ 

1      CO^    CO 

38/56 

CO       CO      1 

111 

37/55 

^2     1        1 

39/57 

CO       1      CO 

40/58 

\      CO      CO 

41/59 

CO      CO      1 

42/60 
1    ft»2    1 

CO       \      Oi 

w'  1   1 

CO      CO       1 

1  ft>2  1 

1      CO      CO 

CO     CO       \ 

CD       CO      1 

CO       X      CO 

G>2     1        1 

CO       CO      1 

1     co^    1 

1       CO      CO 

43/61 

1      CO     CO 

44/62 
1     1    cd" 

45/63 

CO     1      CO 

46/64 
co^    I      1 

47/65 
1    1    ft>2 

48/66 
1     a>'    1 

CO      1      CO 

1      CO     CO 

1     1    co^ 

CO      CO       1 

CO      1      CO 

1       CO      CO 

1     1    a^' 

CO      1      CO 

1      CO     CO 

CO       1      CO 

1      CO      CO 

CO      CO       1 

49/67 

1      CO     CO 

50/68 

CO      CO       1 

51/69 
1    ^2    1 

52/70 
«2   1    1 

53/71 
1     1    co^ 

54/72 
1     a>2    1 

1      1    co^ 

1     co^    1 

CO   a?'  co^ 

1     1   co^ 

1    <e>'    1 

ft>2    1     1 

CO      1     CO 

1       CO      CO 

1     1    co^ 

CO^   CO    co^ 

CO    CO^    0? 

CO^    CO^    CO 

78/99 
0    fu^    0 
10    0 
0    0    1 

84/102 
10  0 
0  10 
0     0  a)2 

90/108 


0    0     1 

0     10 

0     10 

0     0  co"" 

0    0     1 

0    cu^   0 

10    0 

0     0  a>2 

0     0     1 

10     0 

co""   0      0 

0    0     1 

0    w«   0 

1     0    0 

co^  0     0 

0     10 

0     10 

10    0 

III.      Tlie  Hesse   Group  o/"  216    Gollineations. 


23 


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Multiplication  Table  of  Point-  on  Line-Conics  giving  the  collineations  according  to  number. 


24 


III.     The  Hesse  Group  0/  216   CoUineations. 


109/127 
co^  1    1 

110/128 
1      1    w^ 

111/129 
1     co^    1 

112/133 

1       CO       CO 

113/134 
1      1    w^ 

114/135 
1     co'     1 

1        1     (0^ 

1     co^   1 

^02     1       1 

CO^    CO^     CO 

CO       1      CO 

1       CO       CO 

1     <e>2    1 

a>2    1     1 

1        1     CO^ 

1    c6^    1 

0          0 

CO    co~   co^ 

CO^    CO^     CO 

115/130 
1     (0    CO 

116/131 
1      1    co^ 

117/132 
1     ce>2    1 

118/136 
co^    1    1 

119/138 
1      1    co' 

120/137 
1     co^    1 

1     1     co^ 

CO^    CO   co^ 

CO    co^   co^ 

1     <t>2  1 

1       CO     CO 

CO^     CO^     CO 

CO^    CO    co^ 

1       CO     CO 

CO     CO       1 

1     1    w^ 

co^  CO    co'^ 

1        CO       CO 

121/142 

1      CO      CO 

122/143 
1      1    w^ 

123/144 
1    co""    1 

124/139 

1       CO       CO 

125/140 
1     1    co'' 

126/141 
1    co'    1 

CO^   CO    co^ 

CO     CU^    Cl? 

1     1     co^ 

1       W2      1 

co'    1     1 

CO     CO       1 

1      1    co"" 

145/181 
1      1     1 

O)      1       CO 

146/182 
CO    co^    1 

co^    1      1 

CO^    CO^     CO 

1     co'   1 

149/185 
1      CO    co^ 

CO     CO^    CO^ 

150/186 
co"^    CO     1 

147/183 
CO     1     w^ 

148/184 
1      1    1 

CO     1    co^ 

1    1  1 

CO    co^    1 

CO^    CO     1 

1    1  1 

1      CO    co^ 

CO    co^    1 

CO      1    w^ 

1   1  1 

1     CO    co"^ 

co^   CO    1 

1    1  1 

151/187 
111 

152/188 

CO^      1       CO 

153/189 

1       CO^     CO 

154/190 
1      1    1 

155/191 

1       CO^      CO 

156/192 
1      CO    co^ 

1      CO^     CO 

1   1   1 

CO^     1      CO 

CO    co^    1 

1    1    1 

CO^    CO     1 

CO^     1       CO 

1       CO^     CO 

1    1    1 

CO     1    co'^ 

CO^     1       CO 

1    1  1 

157/193 
1      1     1 

158/194 

CO^     CO     1 

159/195 
CO    co^    1 

160/196 
1      1     1 

161/197 
CO     1     co^ 

162/198 

CO^     1       CO 

co"^     1     CO 

111 

CO     1    co^ 

1      CO    co^ 

1    1  1 

1       CO^     CO 

1       CO^     CO 

163/208 
1    co"^   co^ 

1       CO     CO^ 

164/209 

1       1      CO 

1    1  1 

CO^    CO      1 

cu    co^    1 

167/212 
co''   co^    1 

1    1    1 

168/213 

1      CO     1 

165/210 
1      w     1 

166/211 

w      1      1 

1     CO       1 

CO      1       1 

co'    co^    1 

w^    1    co^ 

CO     1       1 

1        1     CO 

1  1  w 

w^   1    co^ 

CO      1        1 

1         1      CO 

1      CO       1 

1    co^   co^ 

169/214 
CO     1      1 

170/215 

1       1       CO 

171/216 

CO^     1      fO^ 

172/199 
1    io''   co' 

173/200 

Cl/    co^    1 

174/201 
co^   1    co^ 

1      CO       1 

1    co^   co"^ 

1        1      CO 

1        1       CO 

1        CO        1 

CO      1     1 

W2    ^2      1 

I       CO      1 

CO       1         1 

1    w     1 

CO       1        1 

1         1      CO 

175/202 
w    1     1 

176/203 

1       1      CO 

177/204 

1       CO      1 

178/205 

CO      1       1 

179/206 
1     1    co' 

180/207 

1       CO        1 

co''    co''    1 

Co'     1       f«>2 

1         W2      ^y2 

1         1      CO 

1      CO      1 

CO       1        1 

1       CO        1 

CO      1       1 

1      1       <«> 

co^    1    co' 

1     co'  co' 

CO^    CO        1 

22.  From  the  system  of  forms  for  the  G210  as  given  by  Maschke  (Math.  Ann. 
33:324)  we  see  that  all  the  collineations  can  be  deduced  from  six  particular  ones 
of  the  conies  in  line-foi'm  with  two  of  these  in  point-form  also.     These  six  are 


Ill,      The  Hesse  Group  0/  216   Collineations. 


25 


Ai  Si ,  A2S1,  J-g  Si ,  AiS-z,  A2  S2 ,  and  Gi  S^ ;  with  the  point-conics  Ai  Si  and 
Ai  S2 .  All  the  216  collineations  can  be  deduced  from  these  conies  and  no  other 
collineations  arise  therefrom. 

The  Tetrahedral  Stjb-Groups. 

23.  The  sub-groups  arising  from  the  isomorphism  with  the  6^13  in  the 
plane  are  these : 

to  the  identical  transformation  corresponds  1  Gi^ , 

"    "   3  sub-groups  G2  correspond  3  G^ss's, 

"    '*    4     "        "        Gs  "  4  G^'s, 

"    "    1     ''       "        Gi  corresponds  1  6^73. 

The  four  flex-triangles  are  regarded  as  the  four  faces  of  the  regular  tetrahedron. 
The  collineations  naturally  then  transform  the  four  triangles  thus : 

The  triangles  AB  CD  are  sent  respectively 

into       "  A  BCD  by  collineations  1-18, 

"         "  BADC  ''  "  19-36, 

"  "  CDAB  ''  "  37-54, 

"  "  DCBA  "  "  55-72. 

The  first  eighteen  form  the  identity  Gi^-,  the  other  three  sets  of  eighteen 
each,  each  with  the  collineations  of  the  Gi^,  form  the  three  G^s  which  inter- 
change the  triangles  by  twos.  All  these  form  the  one  6r^3,  which  permutes 
all  four  triangles. 

Next  the  triangles  are  permuted  as  follows : 

triangles     AB  CD     are  sent  respectively  into  triangles 

AD B  C    by  collineations  numbered  73-90, 


1. 


2. 


3. 


4. 


ACDB 
DBAC 
CBDA 
{  DACB 

\bdca 

CABD 
BCAD 


91-108, 
109-126, 
127-144, 
145-162, 
163-180, 
181-198, 
199-216. 


The  collineations  of  each  of  these  four  sets  in  connection  with  those  of  the  6^13 
form  a  sub-group  G^i ,  leaving  one  triangle  unaltered  and  permuting  the  other 
three  cyclically. 


26  III.     The  Hesse  Group  of  216  OoUineations. 

The  collineations  of  the  Gig  of  course  send  each  respective  inflexional  triangle 

into  itself,  but  not  point  for  point  in  each  case.     A  comparison  of  the  G^^  with 

respect  to  triangle  A  at  once  shows  that,  since  A  is,  therefore,  each  triangle  is 

sent  by  three  collineations  identically  into  itself.     The  collineations  doing  this 

are  in  each  case  identity  and  two  of  those  of  period  three.     Thus  the  action  of 

identity  and  the  eight  of  period  three  is  accounted  for.     The  remaining  nine 

collineations  of  period  two  divide,  differently  for  each  triangle,  into  sets  of  three. 

Those  of  each  set  act  the  same  on  the  triangle  considered,  sending  one  vertex 

into  itself  and  interchanging  the  other  two;  that  is,  in  other  words,  as  is  seen 

by  comparing  the  equations  of  the  harmonic  polars,  one  of  the  three  harmonic 

polars  on  the  fixed  vertex  of  the  triangle  remains,  being  the  fixed  line  of  the 

collineation,  and  the  other  two  harmonic  polars  on  that  vertex  are  sent  the  one 

into  the  other. 

Other  Sub-Groups. 

24.  These  sub-groups  are  those  observed  by  noting  the  effect  of  the 
collineations  on  the  36  conies.  Notice  that  the  conies  naturally  divide  into  sets 
of  nine,  each  set  composed  of  three  sets  of  three. 

By  examining  the  collineations  of  the  G^^  we  readily  see  that  numbers 
1,  4,  5,  10,  11,  12  form  a  dihedral  group  Gq,  sending  conic  B^Si  into  itself. 
Hence  from  symmetry,  there  are  12  dihedral  G^s,  one  for  each  set  of  three 
conies.  These  sub-groups  each  contain  the  identity  collineation,  two  of  period  3 
and  three  of  period  2 ;  there  is  thus  in  each  dihedral  G^  a  cyclic  G^  which  is  an 
invariant  sub-group.  So  the  transform  of  one  of  the  collineations  of  period  two 
by  one  of  the  G^  is  one  of  those  of  period  two. 

The  6^18  contains  identity,  eight  collineations  of  period  3  and  nine  of  period  2. 
The  former  ones  compose  four  cyclic  G^s ;  the  latter  with  identity  compose  nine 
cyclic  G28. 

Next,  we  observe  that  the  collineations  of  the  three  sets  of  18  each  which 
with  the  6^18  ^o^m  the  (r^g,  are  all  of  period  four.  These  by  sixes  have  a  common 
second  power.  See  Valentiner  loc.  cit.  p.  69.  These  nine  second  powers  are  the 
nine  collineations  of  period  two.  So  one  transformation  of  second  order,  the  two 
of  fourth  order  which  produce  it,  and  identity  form  a  sub-group  of  period  four. 
Thus  there  are  twenty-seven  cyclic  G^&.  It  seems  strange  that  Valentiner 
should  have  overlooked  and  even  practically  denied  the  existence  of  the  G^2i6> 
when  he  knew  there  must  be  54  transformations  of  period  four,  and  states  that 
this  number  is  \  N,  where  N  represents  the  total  number  of  transformations. 


III.     The  Hesse   Group  o/  216   Collineations.  27 

He  further  gets  the  (r^g  with  its  sub-groups.  I  am  inclined  to  question  the 
complete  accuracy  of  the  French  resume  for  at  the  end  of  §45,  pp.  173,  227, 
the  Danish  and  the  French  are  exactly  contradictory. 

The  cyclic  G^b  divide  the  conies  into  sets  of  two  and  four,  half  of  a  set 
belonging  to  each  of  two  natural  sets  of  three. 

In  each  of  the  eight  sets  of  18  each,  which  remain  to  be  considered, 
9  collineations  are  of  period  three  and  9  of  period  six.  These  former  of  each  of 
the  four  sets  (1,  2,  3,  4,  p.  25)  of  thirty-six  with  identity  form  9  cyclic  6^3's,  or 
in  all  36  more  cyclic  (rg's  making  the  total  number  40  cyclic  G^n.  These  6^3's 
separate  the  conies  into  sets  of  three  and  permute  them  in  the  set.  Further, 
the  collineations  of  period  six  in  each  set  of  36  appear  by  twos  in  9  cyclic  Gq9,. 
A  cyclic  (rg ,  different  from  a  dihedral  (rg ,  contains  two  collineations  of  period  6, 
two  of  period  3,  one  of  period  2,  and  identity.  There  are  thus  from  each  set  of 
36  collineations  above  mentioned,  together  with  the  9  of  period  two  and  identity 
each  time,  nine  cyclic  G^b  or  in  all  36  cyclic  Gq&.  These  separate  the  conies 
into  sets  of  nine.  Two  of  the  collineations  of  a  G^  send  one  of  the  nine  conies 
into  itself,  two  more  send  the  same  conic  into  each  of  the  other  two  of  its  natural 
set  of  three,  and  the  remaining  two,  which  are  of  period  six,  send  it  into  one  of 
the  remaining  six  conies  of  the  set  of  nine. 

Jordan  says,  loc.  cit.  p.  18,  that  a  group  G  belonging  to  this  (Hessian)  type 
(of  order  24*94))  contains  a  group  H  of  order  27.  "The  substitutions  of  G  are 
permutable  with  ^."  In  each  G^  there  is  such  a  group  6^37  consisting  of  the 
26  collineations  of  period  three  and  identity.     So  further  we  have  4  6^27's. 

Three  of  the  collineations  of  period  three  from  each  lot  of  36  above  referred 
to,  together  with  two  of  the  eight  of  order  three  in  (rig,  and  identity,  form  an 
Abelian  6^9.     Thus  finally  there  are  four  Abelian  6^/s. 

25.  To  recapitulate,  the  Hesse  group  contains  besides  the  sub-groups  of 
§  23,  these  others : 

9  cyclic  G^s,  40  cyclic  6^3's,  27  cyclic  G^s,  12  dihedral  G^s,  36  cyclic  (rg's, 
4  Abelian  G^'s  and  4  G^^b.  These  arrange  themselves  in  interesting  form  on  the 
multiplication  table  as  the  heavy  rulings  help  to  indicate. 

The  classification  as  to  periodicity  is 

1  collineation  identity, 
9  collineations  of  period  2, 


80 

ii 

(I 

(I 

3, 

54 

t( 

11 

ii 

4, 

72 

f( 

11 

11 

6. 

28  IV.     Perspective   Triangles. — Complete  Pappus  Hexagon. 

IV. 

PERSPECTIVE  TRIANGLES.— COMPLETE  PAPPUS  HEXAGON. 

26.  Having  treated  so  fully  the  relations  of  the  four  inflexional  triangles, 
which  are  in  six-fold  perspective,  we  should  show  how  to  obtain  sets  of  triangles 
in  the  other  perspective  forms.  Historically,  it  is  of  interest  to  note  some  papers. 
In  1870,  H.  Schroter  (Math.  Annalen  2:553-562)  set  for  himself  the  question 
whether  a  triangle  can  be  in  more  than  single  perspective  with  another  triangle, 
and  if  so,  in  what  other  forms.  His  paper  is  characteristically  clear  and  easy  of 
reading,  its  method  is  synthetic,  and  it  presents  a  construction  for  triangles  in  all 
possible  forms;  viz.,  one-,  two-,  three-,  four-,  and  six-fold  perspective. 

J.  Valyi,  whose  paper  was  referred  to  in  §  16,  p.  17,  set  for  himself  in  1882 
the  same  problem  and  reached  the  same  results  analytically  without  reference 
to  Schroter.  Some  other  papers  directly  or  indirectly  presenting  perspective 
triangles  are  simply  noted  : 

Rosanes:  Ueber  Dreiecke  in  persp.  Lage.     Math.  Ann.  2:549. 

Hess:  Beitrage  z.  Theorie  d.  mehrfach  persp.  Dreiecke.     Ibid.  28  :167. 

Third:  Triangles  triply  in  Persp.     Proc.  Edinburg.  Math.  Soc.  XIX,  p.  10. 

L.  Klug:  Desmische  Vierseiten-Systeme.     Monatshefte  (1903)  XIV,  s.  74. 

M.  Pasch:  Ueber  Vier-eck  und  seit.     Math.  Ann.  26  :  211-216. 

Caporali:  Memorie,  pp.  236,  252. 

Veronese :  Sull'  Hexagrammum  mysticum.   Lincei  Mem.  II,  1  (1877),  p.  649. 

Triply  Perspective  Triangles  in  Circular  Coordinates. 

27.  It  is  well  known  that  two  concentric  equilateral  triangles  are  triply 
perspective.  We  take  two  such,  first  as  point-triads,  with  coordinates  of  vertices 
resp.,  12  3  1'  2'  3' 

1  o         0)^  at        i^at       oat 

These  two  point-triads  are  in  perspective  thus, 

12  3  having  Center,  and  Axis  of  perspective 

a,  A, 

/?,  B, 

y,  r. 


(17) 


1' 

2' 

3' 

2' 

3' 

1' 

3' 

1' 

2' 

IV.     Perspective  Triangles. —  Gcmiplete  Pappus  Hexagon.  29 

We  name  the  vertices  of  the  triangle  of  axes  opposite  correspondingly  named 
axes  A,  B,  G. 


The  line  11'  has  coord. 

1  — 

a 

'J' 

at—  1, 

"         4. 
^-at, 

(f       ((    00/    '<       '< 

o^- 

aa 

t  ' 

G)^  at  —  6), 

(d^a 

— a  at, 

t 

(t       11    00/    11       11 

0)  — 

t    ' 

0)  a<  —  G)^, 

G>a         0    . 

— cjrat. 

z 

The  determinant  of  these  coordinates  vanishes  identically  for  the  sum  of  each 
column  is  zero.  Therefore  the  lines  meet  in  a  point  a.  Similarly  we  show  the 
points  /?  and  y,  and  find  then  the  coordinates  of  the  three  points  to  be 

''-^~  t{l-a')  '  ^-^-  t{\—a^)  '  r-""-  t{\-a')  '  ^^^^ 
From  these  coordinates  we  see  that  these  three  centers  of  perspective  are  on  a 
circle  concentric  with  the  circles  about  the  given  triads,  equally  spaced  and 
ordered  as  1,  2,  3.     The  radius  of  their  circle  is 


1  — aW 


(at^^\)(a  —  t')  ^jg^ 


With  radii  of  the  given  circumcircles  resp.  a  and  h  this  derived  circle  has  radius 


ah        \  (at^  —  b){a  —  ht^) 
The  coordinates  of  sides  opposite  the  vertices  1,  2,  3  resp.  are  proportional  to 

(1,     1,     1),  (1,    G)^    G)),  (1,    G),    G,^).  (20) 

Those  of  sides  opposite  1',  2',  S',  resp.  are 

(1,  t^,  at),         {1,  (ot^,  cy'at),         (1,  (^H\  (oat).  (21) 

Taking  these  sides  in  the  same  order  of  perspective  as  above  we  deduce  the 
coordinates  of  the  axes.     From  coordinates  (20)  and  (21),  we  write  off  at  once 

_at  —  t^  __(^at  —  GiU"-  _Gi^at  —  c}t^ 

where  x^,  is  the  point  of  intersection  of  sides  i  and  i'. 

These  three  points  lie  on  a  line  for  we  have  identically 

at—t^  1—at  t^—1 

a  at  —  a^t^         a  —  cd^at         ot^  —  g)^     =0. 

Q^at  —  ot^         Gj^  —  a  at         a^t^  —  g) 


30 


IV.     Perspective  Triangles. — Complete  Pappus  Hexagon. 


Thus  we  get  the  coordinates  of  axis 

A  to  be  [a<3_i,         t{a—t%  t^{l  — a^)]. 

B   "   "  [«<'  —  !,         i^H{a—t%         o<2(i_a2)].  (22) 

r   "  "  [«<'  — 1,         i^t{a—t%  GiH^il—a^)-]. 

From  these  we  get  the  coords,  of  the  vertices  of  this  triangle  of  the  axes  thus, 

a,H{a  —  t^)     6)^2(1— a^) 


The  meet  of  BF  is  J.  =  x  = 


at^—1      G>H{a—t^) 
at^—1      at(a  —  t^) 


{l—a^)t' 
at^  —  1 


o  _  G)  (1  —  a^)  t'^ 


_  G>^1 -a^)t' 


(23) 


C  =  x  = 


Similarly, 

'oF—  1      '         ^        ^  otF—  1 

Clearly  these  points  form  an  equilateral  triangle,  concentric  with  the  other  three, 
and  ordered  as  1,  2,  3.     The  radius  of  its  circumcircle  is 


^^—')^(ae-tia- 


{at^—l){a  —  ty 
The  radii  of  the  four  circumcircles  are  seen  to  form  the  proportion. 


(24) 


a         \{al-V^{a-f)_  ,     1 t^ 


{at^-l){a  —  t^) 

or,  in  other  words,  the  derived  circles  are  a  pair  mutually  inverse  as  to  the  same 
circle  as  to  which  the  original  circles  are  inverse.  Therefore,  we  may  state  the 
theorem, 

Two  concentric  equilateral  triangles  are  in  triple  perspective  with  their  centers 
0/  perspective  and  the  three  axes  also  equiangular  triads  concentric  with  the  original 
two;  and  the  radii  of  the  circumcircles  of  the  latter  two  triads  are  functions  of  the 
radii  of  the  original  circumcircles  and  of  the  clinant  of  the  angle  between  the  given  triads. 
The  product  of  the  radii  of  the  latter  two  circles  equals  the  product  of  those  of  the 
former  two;  hence,  the  circles  are  by  pairs  mutually  inverse  in  the  same  circle. 

The  point-triads  1,  2,  3,  and  1',  2',  3',  are  each  in  triple  perspective  with 
a,  ^,  y,  thus, 

12       3         with  perspective  centers         1'      2' 

1' 


a 

7 

P 

/? 

a 

Y 

r 

8 

a 

a 


r 

/3 


/3 

a 
7 


7 
a 


centers 
1 
2 
3 


So,  as  is  known,  two  point-triads  in  triple  perspective  with  their  triad  of  centers 


IV.     Perspective  Triangles. — Complete  Pappus  Hexagon.  31 

of  perspective  are  a  set  of  triads  two  and  two  in  triple  perspective  with  the  points 
of  the  third  triad  as  centers. 

Triangles  1,  2,  3  and  A,  B,  G. 

28.  These  point-triads  as  here  written  are  in  anticyclic  triple  perspective 
with  centers  of  perspective  a',  ^',  y'.  From  coordinates  (17)  and  (23)  we  write 
the  coordinates  of  the  perspecting  rays : 

a—t^-.(l  —  a^)t  1  — q<3-f  (l_a»)f2  t{l  —  a'){t*-{-at^  —  at—l) 

1^=  ^Z^T'  '  at'  —  l  '  {a  —  t'){at'—l) 

-^    ai^{a  —  f)  —  oi{\  —  a^)t     6^(1  — a<^)4- q>^(1— o^)<'^    oit{\  —  a^){t^-\'Oiat—ai—cS) 
^^'  ^r=^F  '  at^—\  '  {a  —  t^){at^—\) 

ai{a^t^)  — 0,^1— a?)t      ai^{l—at^)^ai{l—a?)e     ftj''<(l— a^)(<^  +  tt>^a<— a<— q>») 
^^'  ^"Z::?  '  a<8  — 1  '  [a  —  e){at^—\) 

The  determinant  of  these  coordinates  vanishes  identically,  therefore  the  three 
lines  meet  in  a  point.     From  these  and  similar  coordinates  we  find, 

""  -^-a(a^3— 1)'     ^  -""^  a(at'-l)'     ^  -  ^       a  (a<«  -  1)  *    ^^^^ 


The  radius  of  their  circumcircle  is -*  ~^-— « — ^yt lar*  (26) 

a       \  {af^  —  l)(a  —  T) 

The  axes  of  perspective  are  found  by  taking  the  intersections  of  the  sides  thus, 

12      3 

A     r    B     giving  side  of  triangle  1'  2'  3'     opposite  1'. 

•p  A         p  ((  II        II  H  it  tt  ol 

p        T>  A  (f  H        H  (I  tt  tt  ol 

By  the  same  steps  we  show  that 

Triangles  1',  2',  3',  and  A,  B,  O, 

are  in  cyclic  triple  perspective  with  centers  of  perspective  a",  P",  y",  and  axes 
of  perspective  the  sides  of  triangle  1,  2,  3  in  order  1,  3,  2.  The  radius  of  the 
circumcircle  of  a",  ^",  y"  is 

From  a  comparison  of  the  radii  of  a^y  (19),  of  a'^'y'  (26)  and  of  a"^"y'' 
(27),  we  may  summarize  thus: 

The  circumcircle  of  a,  /?,  y,  which  are  the  perspective  centers  o/|i/  o'  3'P  "^^ 


32  IV.     Perspective  Triangles. —  Complete  Pappus  Hexagon. 

{1   2  31 
A  O  b\^  u'Aoee  axes  are  1',  2',  3',  are  inverse 

to  the  circle  of  1,  2,  S;  also  the  circle  of  a,  (3,  y,  and  th/it  of  a'',  /3",  y",  the  centers 

{V  2'  3'1 
A   T>  n\i  whose  axes  are  1,  3,  2,  are  inverse  to  the  circle  of  1',  2',  3'. 

The  reader  can  very  readily  draw  the  figure  for  all  the  steps  above  outlined, 
so  we  leave  that  to  him. 

29.  As  a  more  interesting  source  of  triply  perspective  triangles,  and  one 
involving  some  considerations  of  the  cubic,  we  present  what  we  shall  call 

THE  COMPLETE  PAPPUS  HEXAGON. 

In  Pappi  AlexAndrini  Mathematicae  Collectiones^  a  Federico  Commandino 
Urhinate  as  Prop.  138,  p.  368,  we  read: 

Si  parallelae  sint  AB,  CD,  atque  in  ipsas  incidant  quaedam  rectae  lineae 
AD,  AF,  BC,  BF,  &  ED,  EC,  jungantur,  rectam  lineam  esse,  quae  per  GMK 
puncta  transit;  and  as  Prop.  139,  p.  368: 

Sed  non  sint  ABCD  parallelae,  &  in  puncto  N  conveniant.  Dico  rursus 
rectam  lineam  esse,  quae  per  GMK  puncta  transit. 

These  theorems  Pappus  proved  by  proportion,  the  equal  ratios  being 
respectively  between  two  lines  and  between  the  rectangles  of  two  pairs  of  lines. 

Salmon^  has  the  same  theorem  stated  thus: 

"If  ABC  are  three  points  of  one  line  and  A!B'0  are  three  points  of  another 
line,  then  the  intersections  BC'IEC,  CA'/CA,  AB/A'B  lie  on  a  line.'' 

The  most  important  mention  of  the  simple  case  is  by  Rudolf  Boger,^  who 
gives  it  as  a  simple  form,  free  from  the  perspective  relations,  of  Das  Sechsech 
in  der  Geometrie  der  Lage. 

30.  The  complete  figure  is  constructed  thus : 

The  numbers  1,  3,  5,  and  2,  4,  6,  are  regarded  as  the  names  of  points  or  of 
lines,  each  set  of  three  lying  on  a  line  or  a  point  resp.  We  consider  the  cross- 
joins  as  follows: 


1 

i   2 

1  3 

> 
1  4 

1  6 

Ui 

(1) 

(2) 

(3) 

'1' 

3 

2 

5 

4 

1 

e 

(4) 

(6) 

(6) 

[2] 

5 

2 

1 

4 

3 

6 

(7) 

(8) 

(9) 

[3] 

3 

2 

1 

4 

5 

6 

(10) 

(11) 

(12) 

V 

5 

2 

3 

4 

1 

6 

(13) 

(14) 

(15) 

'6' 

1 

2 

5 

4 

3 

6 

(16) 

(17) 

(18) 

[«] 

on  2i  or 
Di  resp. 

on  ^2  or 
Z>2  resp. 


(28) 
(29) 


iProm  U.  8.  Cong.  Library;  also  by  F,  Hnltsch  In  3  Vols,  giving  the  Greek  text  also,  see  Vol.  2,  p.  885. 

«  Conic  Sections  (6th  Ed.)  §268,  p.  246,  Kx.  1. 

*Sechseck  and  Involution,  Mitteilangen  d.  Math.  Ges.  in  Hambarg,  Bd.  Ill,  Feb.  1899,  s.  887. 


IV.     Perspective  Triangles. —  Complete  Pappus  Hexagon.  38 

This  means,  when  the  six  numbers  are  points ^  that  the  line  of  1,  2  intersects 
that  of  4,  5  in  the  point  (l);  the  line  2,  3  cuts  line  6,  6  in  point  (2);  and  3,4/6,1 
is  the  point  (3);  further,  that  point  (1),  (2),  (3)  lie  on  a  line  [1]  which  is  one  of 
three  lines  similarly  got  and  lying  on  point  2i .  Observe  there  are  nine  lines, 
such  as  1,  2,  connecting  the  two  sets  of  three  points  each,  which  have  not  been 
named.     In  the  dual  figure  we  shall  call  the  nine  points  ai_3,  /?i_3,  71-3. 

On  the  figures,  the  six  points  shall  be  marked  Tt^  and  the  six  lines  P^ ;  the 
points  or  lines  (n)  shall  be  marked  simply  n.  For  convenience  in  the  analytical 
work  slight  changes  are  there  made  as  to  names  but  these  will  be  readily  followed 
on  the  figures  throughout.  Following  the  names  given  to  points  and  lines  in  the 
Pascal  hexagon,  the  points  2i  and  ^2  are  called  Steiner  points,  whose  line  is  S) 
the  lines  P^  are  called  Pappus  lines ;  and  we  call  the  lines  Di  and  D^  Hessian 
diagonals,  which  intersect  in  h. 

31.  As  a  convenient  projection  of  the  hexagon,  we  take  the  points  nz^rii^y 
Ttg  on  a  line  considered  the  axis  of  reals,  and  the  points  Ttj,  Tta,  7^5  on  the  line  at 
infinity  so  that  the  lines  from  the  three  points  on  the  axis  to  these  three  are 
equispaced  lines,  parallel  respectively 

to  x-=.ty  x^iidty  x^=-{^ty 

going  resp.  to        n^  Ttg  itf, 

Lines  on  ;r2  =  a:  a!=<y — a{t — 1).      x-==.(oty — a{ii)t  —  1).     xz^to^ty — a{a)^t — 1). 

«       "  714  =  6:  x=zty—b{t—l).      x  =  ioty—h{oit—l).     x=za)Hy—h{o}H—\),     (30) 
"      "  7Z'6  =  c:   x=zty — c{t — 1).      x=zu)ty — c{a)t  —  1).     x:=(o^ty — c{a)^t — 1). 

The  points  (p)  — 1>  =  1,  2, ,  9  —  are  in  general  thus, 

.,,            .            a—ba  —  (a—b)o>*+U 
Tti^a  With  % b  IS  x  r= -^— 1 ; 

and  the  points  {q)  —  g-  =  10,  11,  .'...,  18  —  of  § 30  are  thus, 

rti,o  with  %6  is  a;  = ^  _   3    ' ; 

or  points  {q)  are  got  from  points  {p)  by  interchanging  b  and  c  in  the  equation 
where  p-=.q—  9. 

In  the  above,  a  and  b  each  permute  for  a,  &,  c,  but  a  is  never  b ;  and  t,  j 
are  each  1,  3,  6,  but  i  is  never  j  in  any  one  equation,  a,  5,  c  are  the  general 
points  Ttg,  7t4,  Tte  along  line  D^. 

These  18  points  in  6  sets  of  three  each,  as  indicated  in  §30,  lie  on  six  lines  P^. 
From  the  equations  of  points  (^)  and  {q)  we  get  the  coordinates  of  the  lines  P. 


34  IV.     Perspective  Triangles. — Complete  Pappus  Hexagon. 

For  lines  Pi,  where  i  =  1,  3,  6,  we  find  coordinates  are 
P,:  a-{-hQ)-\-ooi^,      — {a-^ha}^-\-co})or*tf       o)'^{ab-{-bca)-\-caaP) — {ab-^hca)^-{-ca(o)or*^H. 

The  coordinates  of  lines  Pj,  where  y=  2,  4,  6,  are 
Pj:  a-\-ho?-\-ciOy     — {a-\-bo)-^cii?)ari^H,      a) {ah -\-hcu)^ ■\- caio) — {ah-\-bcco-\-caa)^)arH. 

The  three  lines  P<  lie  on  the  point  2i ,  which  is 


X=.  — 6) 


ah  -f  &CO  H-  cao)^ 
a  -\-  hdi  ■\-  cu)^ 

and  the  three  lines  P,  lie  on  the  point  Sg,  which  is 

a  +  &w^  +  CO) 


jc  =  — a 


These  two  points  are  evidently  conjugate  and  therefore  symmetrical  as  to 
the  axis  of  reals.  Further,  since  the  axis  of  reals,  or  Z>2,  bisects  the  line  between 
Si  and  Sg,  the  two  points  2  are  harmonic  as  to  the  two  lines  D.  They  are  the 
Hessian  pair  of  the  three  points  Ttg,  714,  Ttg. 

Since  the  points  2  are  independent  of  t,  these  points  remain  the  same  for 
any  three  equispaced  lines  on  a,  h,  c  resp.,  mutually  parallel ;  or  keeping  one 
triad  of  points  fixed  the  triad  on  the  other  line  may  move  all  along  their  line 
subject  only  to  the  condition  that  the  angles  between  the  lines  on  the  fixed  points 
remain  constant.  That  is,  2i  and  ^2  are  the  same  for  all  triads  on  the  second  line 
having  the  same  Hessian  pair.  Thus  along  the  one  line  may  be  generated  a 
pencil  of  triads  with  the  same  Hessian  pair  by  turning  the  equispaced  triad  on  X, 
each  three  points  cut  out  at  any  instant  being  a  triad  of  the  pencil. 

Reversing  the  Process. 

32.  Starting  with  the  three  lines  P<  on  2i  and  the  three  Pj  on  Sg,  the  lines 
Pij  intersect  in  nine  points  as  follows : 

n  j3  ,      CL^  —  he  -\-  (a  —  h){a  —  6)0^  t 

2a  —  0  —  c 

p  D  .  ^^(^  —  ci^  +  {o  —  a){c  —  b)G)t 
Ac  —  a  —  o 

p  p  .  ^^  h^  —  ca  -{-  (b  —  c)(h—  a)t 
26  —  c  —  a 


IV.     Perspective   Triangles. —  Complete  Pappus  Hexagon. 


35 


The  remaining  six  may  be  written  at  once  by  comparing  these  three  with  the 
following  table : 

Pi  P,  P, 


Pi 


a\  o" 


x\  1 


.2 


W,  1  c",  o"         a",  0) 

The  lines  joining  these  intersections  in  pairs  as  indicated  in  §30  meet  by 
threes  in  points  along  D^  and  Z^j,  which  are  not  in  general  the  original  points  on 
these  lines. 

The  lines  {q)  to  the  three  new  points  along  B^  are  given  by  the  second  lot  of 
three  permutations  (29)  and  have  slopes  respectively,  ^^,  w^^,  o^^^;  so  they 
turn  just  twice  the  angle  from  the  axis  as  the  original  pair  and  are  likewise 
equispaced. 

The  lines  (^)  intersect  by  threes  on  three  new  points,  TtJ,  Tts,  Tt^,  along  D^,, 
corresponding  respectively  with  a,  h,  c,  in  order  along  the  external  segment  of 
the  line.     They  are, 

a  —  h         W  —  ca 


x„  = 


—  a         (? — ah 


,  where  a,  6,  c,  permute  cyclically. 


a  —  h         2b  —  c  —  a 
c  —  a         2  c  —  a  —  b 
33.     Thus  far  the  origin  on  the  axis  has  been  arbitrary.     Now  consider  it 
the  centroid  of  the  three  given  points,  so  that 

a  +  6  +  c  =  0, 
whence  also  a^  —  bc=.b^  —  ca-=:.  c?  —  ab 

and  be  -\-  ca  -\-  ab  =^  —  X,     2  a  —  b- 

The  three  new  points,  TtJ,  then  become  resp., 
a'k  b'k 


=  \  say, 
c  =  3a,  etc. 


X  =■ 


X  := 


2bc  -\-  ca  -\-  ah'         *"       he  —  2 ca  +  a5 ' 
The  counter-triad  of  the  three  points  a,  6,  c,  is 
—  2bc  -\-  ca  i-  ah 


X  =■ 


cX 


be  -)r  ca  —  2  a6  * 


be 


3a 
2ca  -{•  ah 


36 
be  -\-  ca  —  2ab 


3c 


from  (xa/bc)  =:  —  1 ;  call  it  a'. 
from  {xh/ca)  =  —  1 ;  call  it  h'. 
from  (xc/ab)  =  —  1 ;  call  it  c'. 


The  cubic  along  the  line  a,  5,  c  is 

x^  —  Xx  —  abe  =  0. 


36  IV.     Perspective  Triangles. — Complete  Pappus  Hexagon. 

Differentiating  this  as  to  a;,  we  have 

3  a:^  —  X  =  0, 
the  roots  of  which  are  the  polar  pair  of  infinity,  the  intersection  of  D^  and  D^. 
Calling  the  roots /and/',  we  have 

/  +  /'  =  0,and//'=-/2  =  -/'«=_|.     ,./2  3^/'2  =  |. 

Then  by  comparing  the  above  values  of  this  paragraph,  we  see 

n[a'  =  n',h'  =  nid=\; 

so  the  new  triad  and  the  counter-triad  of  the  original  triad  are  in  an  involution 
whose  double-points  are  the  polar  pair  of  the  intersection  of  the  lines  Di  and  D^ 
as  to  the  original  triad. 

Further,  starting  with  the  new  triad  as  we  did  with  the  triad  Ttj,  Tta,  7t^^ 
we  shall  secure  a  third  triad  nl^  Ttg,  ni,  etc.  continuously,  all  these  triads  having 
the  same  polar  pair  with  respect  to  h  and  the  same  Hessian  pair. 

Thus,  there  is  constructed,  as  it  were,  a  syzygetic  pencil  of  triads  of  points 
along  a  line.  For  all  the  above  there  is  of  course  its  dual,  giving  a  corresponding 
pencil  of  line  triads  on  each  of  two  points. 

34.  From  the  Complete  Pappus  Hexagon  we  have  the  following  theorems 
with  their  duals : 

I. 

Three  lines  P^  on  each  of  two  points         Three  points  Tt^  on  each  of  two  lines 

2i,  ^2,  joined  by  the  line  S,  intersect  Z^j,  D^,  meeting    in   the  point  5,   are 

cross-wise   in   nine    points   ai_3,  ft_3,  cross-joined  by  nine  lines  which  meet 

yi_3,  which  join  by  18  lines,  which  are  in  eighteen  points,  1-18,  which  are  the 

the  sides  of  two  sets  of  three  point-  vertices  of  two  sets  of  three  line-triads 

triads  each.  each. 

The  triangles  of  each  set  are  inter  se        The  triangles  of  each  set  are  inter  se 

in  triple  perspective,  having  as  centers  in  triple  perspective,  having  as  axes  of 

of  perspective  the  points  2i  and  Sg  each  perspective  the  lines  D^  and  D^  each 

three  times,  and  three  points  Tt^  on  the  three  times,  and  three  lines  P^  on  the 

line  D2  or  D^  resp.;  and  having  as  axes  point  X^  or  2i  respectively;  and  having 

of  perspective  the  line  D^  or  D^  three  as  centers  of  perspective  the  point  2i  or 

times  each  for  the  sets  resp.,  and  twelve  ^2  three  times  each  for  the  sets  resp., 

other  lines  all  of  which  pass  through  a  and  twelve  other  points  all  of  which 

point  e,  which  is  the  pole  of  the  line  S  lie  on  a  line  E^  which  is  the  polar  of 

as  to  any  of  the  six  triangles.     The  18  the  point  h  as  to  any  of  the  six  triangles, 

lines  above  lie  by  three  on  six  points  The  18  points  lie  by  three  on  six  lines 

Ttj,  which  are  three  and  three  on  the  P^  which  are  three  and  three  on  the 

lines  Z>i  and  D^  above.  points  2i  and  Sg  above. 


IV.     Perspective  Triangles. — Complete  Pappus  Hexagon.  37 

II 
The  lines  Z^j,  D2  and  S  are  the  false         The  points  2i,  ^2  ^^^  ^  ^^^  the  false 
sides  of  the  complete  quadrilateral  of    vertices  of  the  complete  quadrangle  of 
the  Hessian  pairs  of  the  line-triads  P^     the  Hessian  pairs  of  the  point-triads 
on  2i  and  on  ^2  •  Tt^  on  D^  and  on  D^ . 

From  these  follows  the  general  theorem,  as  also  from  the  demonstration 

of  §§31,  32. 

III. 

Three  lines  on  each  of  two  points  give  rise  to  three  points  on  each  of  two 
lines,  and  the  latter  by  reciprocating  the  process  give  rise  to  three  lines  on  each 
of  the  original  two  points.  The  derived  three  lines  have  the  same  Hessian  pair, 
or  are  inclined  to  each  other  at  the  same  angle  as  the  original  three. 

35.  We  now  give  a  proof  of  the  theorems  on  the  left.  Take  the  two 
Steiner  points  2i ,  ^2,  to  have  coordinates  0*1,  Cg,  Cs  and  $1,  $3,  $3  resp.;  the 
triangle  aia2a3,  formed  by  the  intersections  of  the  Pappus  lines  as  will  be 
shown,  as  reference  triangle;  and  the  line  S  as  auxiliary  line. 

The  line  S  determined  by  2i  and  X2  is  given  by 

=:  0,   and  will  be  written 

SiXi  -f  ^3X2  +  S^Xs  =  0. 
Since  this  line  is  taken  as  auxiliary  line 

^1=^^2=^3=1;        O-i  -f  (Ta  +  (Tg  =  0,         $i  +  $2  +  $3  =  0,  /g-j^x 

also  $?  Cj  Gk  —  5V  S/fc  (J  i  =  C*  5"*  —  cij  Sj , 

where  i, /,  h  are  each  1,  2,  3  successively. 

The  equations  of  the  Pappus  lines  are  the  corresponding  minors  as  re- 
presented thus, 

P\        P^        Pb  Pz        Pj        Pfi 

^1  <^2  <^3  il  $2  $3 

Xi  X2  2^3  Xi  X^  Xg 

Their  respective  intersections,  corresponding  with  the  nine  points  P^j  of 
§  32,  have  coordinates : 

ai:   (1,0,0),                   a^:  (0,  1,0),  a,:   (0,0,  1), 

A:  (o'lSi,  o'i$2,  <^3$i),     ft:   (<^i52>  0'2$2j  <^2  53),  ft:  ((^sSi,  <Ja5'3,  0'3$3),     (32) 

r  1  ■     (<^l  5l  ,  0-2  5l ,  (^1  $3),       72  '    (<^2  5l ,  (^2^2,  Cr3  $2),         ^3  '     (<^1  ^3 ,  <5'3  $2  >  <^3  $3)' 


a-i 

JC2 

Xs 

O"! 

<^2 

^3 

$1 

52 

$3 

meet  lines 

p. 

(I 

ti 

p. 

11 

n 

p 

38  IV.     Perspective  Triangles. — Complete  Pappus  Hexagon. 

where  the  intersections  are  named  as  indicated  here : 
Lines  P3     P^     P^ 

P2     P4  resp.  in  points  ai     02    ag 

Pi     Pe      ''      "        ''       ft     A     ft  (33) 

-   A   A    "    "     "     73   73   n 

From  coordinates  (32)  we  write  the  coordinates  of  the  joins: 
a^asi   (1,   0,  0).  agtti:  (0,   1,  0),  ajag:   (0,  0,   l). 

ftft:  (t^aSa,  0-353,  <J2$8)-    ftft :  (t^sSs,  OgSi,  (TiSi).     ftft :  (CgS'g,  o-jSi,  <yi52)-    (34) 

7273-     (<y3$2,  0'8$3,  Og^g)-       73/1:     {(^3^3,  ^j^g,  (Tj  $i).       J/j /g  I     (CgSg,  <yi$l,  <y25'l)- 

By  observing  that  the  determinant  of  the  three  sets  of  coordinates  of  each 
column  above  vanishes  identically,  after  subtracting  its  third  row  from  its  second 
row  and  remembering  (31),  we  see  that  the  three  lines  of  each  column  pass  through 
a  point.     These  three  points  are  called  7ti,  Ttg,  715,  and  have  coordinates  resp., 

(0,    — (72  52,    <^3  53),        (<yi$l,    0,    — (Tggg),        (— CTj  $i ,    (Tg  $2,     0)? 

and  these  three  points  clearly  lie  on  a  line,  Z^j,  with  coordinates, 

(<y2<y3$2$3,        (y-sf^lisiu        0'l<y2  5l5'2)    or    (-— ,       -— ,        -~       \  (35) 

N"l  Si  "2  $8  "3  b  3/ 

Again,  from  (32)  we  write  the  coordinates  of  the  other  set  of  joins  thus : 
«ii9i:  (0,  — <T,Ci,  <TiC2).       ^iTi'  (''sCs— ''zCa,  — ^^iCi,  ^^iCi).       ri«i*  (0,  — ''iCs,  <'2fi)- 

"2^:    (''aCs,    0,    — <TiC2).         Ar?:    (<^3C2,    <^lCl— <T8C3,    — <'2C2).         r2«2:    (<^3C2,    0,    — ^TgCi).  (36) 

a^^s:  ( — <T2C8,  ^^aCi,  0).       /JsTj:   ( — ^^sCs,  <"fs,  <'2C2— ^'iCi)-       Ts^s-  ( — <^8<^2>  '^iCs,  0). 

Evidently  the  three  lines  of  each  column  here  also  lie  on  a  point,  for  the 
columns  of  determinants  vanish  identically  on  addition.  The  three  points  are 
called  Ttg,  7ti,  Ttg  resp,  and  have  coordinates, 

^i'  (<'8f2»  ''iCs,  ^^aCi),     TTi:  (<T2C2  +  <'8C3  —  ''iCi,  ^'sCg  +  ^'iCi  —  <'2C3,   <yiCi  +  <'2C2  —  ''sCa), 

^Te:  (tTjCs,  ^TgCi,  0"iC2)« 

These  three  points,  on  adding  columns  of  their  determinant,  are  seen  to  lie  on 
the  line  D^ .     Its  coordinates  are 

The  intersection  of  A  and  2)2,  viz.,  S,  has  coordinates 

[<'l  fl  {^Ti  C2  +  ''S  Cs  —  2  <Ti  Cl),    <T2  C2  (<^8  Cs  4-  ^'l  Cl 2  (Ti  C;;),    ffg  Cs  (''l  Cl  +  ^'s  ^2 2  (Tj  Ci)].       (38) 

The  nine  joins  of  each  of  the  two  sets,  (34)  and  (36),  are  the  sides  of  three 
triangles.     These  are  resp.  the  point-triads 

(1)  tti     ttg     a,         and         (l)     ai     ft     yi 

(2)  ft     ft     ft  "  (2)     02     ft     72 

(3)        71        72        78  "  (3)        Og        ft        7g 


IV.     Perspective  Triangles. —  Gom/plete  Pappus  Hexagon.  39 

whose  vertices  as  shown  in  (33)  are,  for  each  triangle,  joins  two  and  two  of  all 
six  lines  on  2i,  Sgj  and  so  the  triangles  are  cubics  passing  through  the  same 
nine  points. 

The  triangles  of  each  set  are  mutually  in  triple  perspective.  The  three 
centers  of  perspective  are  first  Tt^,  n^,  tIq,  as  the  coordinates  of  lines  (34)  show, 
and  Xi,  X2  each  three  times  as  seen  from  the  naming  of  points  in  (28)  and  (29). 
The  centers  for  the  second  set  are  similarly  Ttj,  n^,  n^,  and  2^,  ^2  each 
three  times. 

The  axes  of  perspective  are  clearly,  once  for  each  two  triangles  of  the  first 
set,  the  line  D^ ,  and  once  for  each  two  of  the  second  set,  the  line  Z^g .  This 
accounts  for  six  of  the  eighteen  axes.     The  remaining  twelve  are  as  follows : 

^   ^    u    00      u      u      u     )n'        ^'    \      Whence,  Axis  I  has  coordinates 
aa«:     "    M.      "      "      "      {-r^,0,C,).  (<'^.  "a, 'i), 

for  by  subtracting  second  row  of  the  determinant  from  the  first  row,  and  the 
third  from  the  second,  we  have 

111 
1       1     1 
— (Tj      0      (Ta 
ttittg  and  /?3/?i  meet  in  point  ($1,  — $3,  0). 
aga3    ''    ftft      '^      ''       ''      (0,S2,-Si). 


(TgCTi 


=  0, 
Whence  Axis  II  has  coordinates 


a,a,    "    13,13,      "     "      "     (-;,,  0,  fs).  fe,  Si,  «.)• 

Thus  in  order  as  indicated,  we  have  the  triply  perspective  triangles : 
Triangles.  Centers.  Axes  of  Perspective. 

8  3/3    J  ^'  ^^'   ^^*  ^^'   ^^^'  ^^'  ^^^'   ^^^'  ^^'   ^^^* 

tti  ttg  ag 

Aft  ft 
7\  7%  rs 

tti  /?i  yi 

"2  ft  /2 

a2ftr3 
asftya 

0^3  ft /3     1  >r         'e  T\         I  \       I  \ 

aifty       J  ^^'    -^IJ    -^2-  ^^1     \P\^    0'3j    <^2),     Ul,     $3,    $2)- 


\              Tti,  Si,  Sg.  A,  ($2;    $3,    $1),     (<^3,    <^1,  ^%i' 

fk,  ^2,  Si.  A,  {<fi,  ^2,   <Ja),   (Si,   ^2,  53)- 

7«1,  Sg,  Si-  A,  (0*2;    <^1,    <^3),     {^2,    Si,  is)' 

7*3,  ^2;  2i.  A>  fo,    $2,    $1),    (<^3,    <^2;  ^^l)- 


40  IV.     Perspective  Triangles. —  Complete  Pappus  Hexagon. 

The  twelve  axes  other  than  2>i  and  Dg  are  seen  to  pass  through  the  point 
with  coordinates  (1,  1,  1),  which  is  thus  the  auxiliary  point  e,  the  pole  of  aS'  as 
to  the  reference  triangle.     Since 

S  has  the  same  pole,  e,  as  to  all  point-  h  has  the  same  polar,  E,  as  to  all  line- 
cubics  consisting  of  three  joins  of  the  cubics  consisting  of  three  joins  of  the 
six  lines  P^,  two  and  two,  six  points  7t(,  two  and  two,   (Salmon: 

H.  P.  C,  §166,  pp.143,  144), 

e  is  the  pole  of  S  as  to  every  and  any  E  is  the  polar  of  h  as  to  every  and  any 
of  the  six  triangles  on  all  six  lines  P^.      of  the  six  triangles  on  all  six  points  Ttj. 

The  18  lines  have  already  been  shown,  (34)  and  (36),  to  lie  by  threes  on  the 
six  points  n^  which  are  on  the  lines  D^  and  Z>g  three  and  three. 

The  lines  A  and  D^  as  diagonals  or  false  sides  of  the  complete 

QUADRILATERAL    OF    THE    HeSSIAN    PAIRS. 

36.    The  following  linear  relation  exists  as  to  the  three  lines  P^  on  2i, 

Cl  ((Tg  iCg  — (T3  X^)  -f  (Tg  (Cg  X^  — (Ti  ag)   +  (Tg  ((Ti  Xg  —<^2^l)=^' 

The  Hessian  covariant  of  the  binary  cubic  is  given  by  the  sum  of  the  squares  of 
these  terms  separately,  and  the  imaginary  Hessian  lines  are 

(Ti  ((Tg  a^g  — (Tg  Xg)  +  6)  CTg  ((Tg  X^  — (Tj  X^  +  id^  (Tg  ((Ti  X^,  — (Tg  Xj)  =  0, 

and  ffi  ((Tg  OTg  — (Tg  arg)  +  6)^  (Tg  ((Tg  x^  — (Tj  iCg)  +  6)  (Tg  {a^  x^  — (Tg  Xi)  =  0. 

These  two  and  the  analogous  two  on  2g  may  be  written  in  reduced  form 

respectively 

(Tg(Tga:i +  6)(Tg(Tia:g  + (o^(TiCrgiCg=0,      ]l) 

(^zO^Xi  -f  G)^(Tgcria:2  -f  cj(Ti(TgiCg  =  0,     {  2[ 

Sggga-i  +  cjSggiiCg-f  (j2$i$2a;3  =  0,        j3f 

$g5'3iBi  +  a)2$g$iiCg  +  G)$i$2a:3  =  0,        J4[ 

These  intersect  as  follows : 

|1}  and  1 3}  in  imaginary  point  J:  (<TiCi,  o>^<J'i<:%,  (oa^z^). 

{2|     «    j4}  "  "  "     J'.{a,<:,,oya^z^,to''a,<:^). 

\1\     "     {4}"  "  "    IT:  (<riCi[<'2C8 -0X^3 C2],  <'2 face's Ci-tt'^^ifs],  <^8C8KC2-w<y2Ci]). 

|2|     "     {3|   "  "  "     £':  (<Tifi[<T2C3-w''<y8C2],  <'2C2[<'8Ci-o>^<'if3], '^sCsL'^iCz-^^aCi]). 

The  line  7/  is  the  real  line  D^  given  by  (35). 

The  line  -£?^  is  the  real  line  D^  given  by  (37). 
Therefore,  D^  and  Dg  are  the  diagonals  of  the  imaginary  quadrilateral  of  the 
Hessian  pairs  of  the  line-triads  on  2i  and  Sg. 


Hexagon  of   three  equispaced  lines  on  each  of  two  points.     Theorem  VI. 


Hexagon  of  three  points  on  each  of  two  lines.     Theorem  I. 


42  IV.     Perspective   Triangles. — Complete  Pappus  Hexagon. 

Theorem  III.  follows  without  further  proof. 

37.     As  may  be  seen  from  Fig.  3,  for  the  reverse  or  dual  process  the  three 
line-triads  of  a  set  are  mutually  in  triple  perspective  thus : 


Triang^ 

les. 

1      6 

8 

2      4 

9 

1      6 

8 

7      3 

5 

2      4 

9 

5      7 

3 

10     14 

18 

16      11 

15 

11      15 

16 

17      12 

13 

10      14 

18 

12      13 

17 

Centers. 

Axes  of  Perspective. 

-1,      «, 

h. 

p., 

A, 

A- 

c,      2i, 

d. 

A, 

A, 

A- 

e,     f, 

2i. 

A, 

A, 

A- 

9,        ^2, 

h.    . 

A, 

A, 

A- 

^)        ^2) 

I. 

A, 

A, 

A- 

^2,     m, 

n. 

.  A, 

A, 

A- 

The  points  a,  h,  c,  ....  w,  ?i  are  the  twelve  points  on  E. 

38.     Without   further  proof  because   they  follow  analytically  from  data 
already  given  and  may  be  tested  in  Fig.  3,  we  present  several  theorems: 

,IV. 

A  triangle  of  one  set  of  three  is  in  two-fold  perspective  with  any  one  of  the 
opposite  set  of  three  triangles,  but  for  all  such  perspectives — 

there  are  only  nine  axes  each  taken  thereareonly  nine  centers  each  taken 
twice  and  situated  on  the  covariant  twice  and  situated  on  the  covariant 
point  e.     The  centers  in  each  case  are     line  E .     The    axes   in    each   case   are 


The  point-  and  line-triads  are  between  themselves  in  single  perspective. 
The  center  of  perspective  in  each  case  is  1.^  if  the  two  triads  are  both  of  the  first 
or  both  of  the  second  set  of  three  as  herein  classified,  and  Sg  is  center  if  they 
are  of  oppositely  named  sets. 


IV.     Perspective  Triangles. — Complete  Pappus  Hexagon.  43 

39.  The  special  forms  or  arrangements  for  the  three  lines  on  each  of  two 
points  to  which  attention  is  called  are, 

(1)  Two  sets  of  equispaced  triads;  i.  e.,  lines  at  angles  of-^. 

(a)  The  points  being  the  two  equiangular  points  of  the  triangles 
of  one  set  of  three. 

{h)  The  points  being  on  the  circumcircle  of  equiangular  triads. 

(2)  The  points  taken  at  infinity, 

(a)  arbitrarily,  giving  two  sets  of  three  parallel  lines  each,  at  an 
angle  3^  with  each  other. 

(6)  at  the  circular  imaginary  points  of  the  plane,  giving  two  sets 
of  perpendicular  lines. 

These  various  forms  are  handled  analytically  best  by  using  special  coordinate 
systems  for  the  several  cases,  and  furnish  nice  work  in  devising  expeditious 
methods.    I  omit  as  irrelevant  the  various  methods  that  were  employed. 

40.  From  these  special  forms  we  have  the  theorems, 

VI. 

For  a  triad  of  equispaced  lines  on  each  of  two  points,  one  set  of  the  three 
point-triads  consists  of  equiangular  triangles  with  sides  respectively  parallel 
(Fig.  4).  Thus  one  of  the  lines  D  is  at  infinity  and  the  other  is  perpendicular 
bisector  of  the  line  S  between  Sj  and  ^2.  Further,  the  circumcircles  of  the 
three  equilateral  triads  pass  through  Si  and  Sg,  and  those  of  the  other  set  of 
three  triangles  intersect  in  e,  the  pole  of  S  as  to  any  of  the  six  triangles. 

VII. 

If  to  the  conditions  of  the  previous  theorem  we  add  that  one  of  the  set  of 
scalene  triangles  is  also  equiangular,  then  the  vertices  of  the  other  two  triangles 
of  its  set  are  inverse  points  as  to  its  circumcircle,  and  all  the  circumcenters  of 
the  set  of  three  equilateral  triangles  are  on  the  finite  Hessian  diagonal  D. 

Theproof  of  this  last  theorem  is  especially  neat  by  use  of  circular  coordinates 


Finis. 


VITA. 

Charles  Clayton  Grove  was  born  December  19th,  1876,  the  son  of  Lewis 
and  S.  Elizabeth  Grove,  at  Hanover,  Pa.  His  early  education  was  received 
in  the  private  school  of  Miss  Martha  E.  Grove  and  in  the  public  schools  of 
Hanover. 

In  1896,  he  was  graduated  from  the  Millersville  (Pa.)  State  Normal  School, 
where  he  prepared  for  college.  After  two  years  of  teaching,  he  entered  in 
September,  1898,  upon  the  classical  course  of  Pennsylvania  College,  Gettysburg, 
receiving  the  degree  of  Bachelor  of  Arts  in  1900.  The  next  winter  he  was 
Supervising  Principal  of  the  schools  of  Hummelstown,  Pa.  In  1903,  the  degree 
of  A.  M.  was  conferred  at  Gettysburg. 

In  October,  1901,  he  entered  upon  graduate  study  at  the  Johns  Hopkins 
University  with  Mathematics,  Physics  and  Italian  as  subjects.  At  its  very  close, 
this  course  was  interrupted  by  his  taking  an  instructorship  in  Mathematics  in 
Pennsylvania  State  College.  During  the  winter  of  1905-6,  he  was  an  instructor 
at  the  Baltimore  Polytechnic  Institute,  and  completed  the  work  for  his  doctorate 
at  the  Johns  Hopkins  University. 


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